MAJORITY DECODABLE CODES
DERIVED FROM FINITE GEOMETRIES
by
K. J. C. Smith
University of North Carolina
Institute of Statistics Mimeo Series No. 5'1
December 1967
This research was supported in part by the National Science
Foundation Grant No. GP-5790 and the Army Research Office, Durham
Grant No. DA-ARD-D-3l-124-G9l0.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
ii
TABLE OF CONTENTS
Chapter
I
II
•
Page
ACKNOWLEDGMENTS
iii
ABSTRACT
iv
SUMMARY
v
INTRODUCTION
1
1.1 The coding problem
1
1.2 Majority Decoding
3
mCIDENCE MATRICES IN FINITE GEOMETRIES
8
2.1
8
Introduction
2.2 Rank of an incidence matrix
8
2.3
10
Finite fields
2.4 Finite projective geometry PG(t,q)
13
2.5
Finite affine geometry EG(t,q)
17
2.6
Incidence matrices of points and (t-l)-flats
21
2.7
Incidence matrices of points and d-flats in PG(t,q)
36
2.8
Incidence matrices of points and d-flats in EG(t,q)
54
III GEOMETRIC CODES
•
61
3.1
Introduction
61
3.2
Cyclic codes
61
3.3 Majority decoding
67
3.4 Geometric codes
72
3.5 Non-primitive geometric codes
86
3.6 Majority decoding of geometric codes
92
BIBLIOORAPHY
105
iii
ACKNOWLEDGEMEN'rS"
I gratefully acknowledge the guidance and encouragement given
me by my adviser, Professor R. C. Bose, during my research.
It is
a priviledge to have attended his stimulating lectures and to study
under him in the area of combinatorial mathematics.
I wish also to thank the other members of the faculty of the
Department of Statistics Who have contributed to
my
graduate training,
in particular Professors 1. M. Chakravatri, N. L. Johnson and M. R.
Leadbetter.
Thanks are also extended to Professor R. R. Kuebler, Jr.,
for his valuable comments on improving this dissertation and to
Professor R. J. Troyer, the other member of
my
graduate committee,
For financial support during my graduate training, I sincerely
thank the Department of Statistics, the National Science Foundation
and the U. S. Army Research Office, Durham.
I gratefully express
my
To these institutions
appreciation.
I would like to thank Professor D. K. Dale of the Department of
Mathematics at Carleton University, Canada, for his encouragement and
stimulation of
my
interest in mathematics and statistics during
my
undergraduate studies.
For her careful typing of the final manuscript I extend
to Mrs. Margaret Laughlin •
•
my
thanks
iv
ABSTRACT
A number of q-ary cyclic error-correcting codes to which a
majority decoding algorithm is applicable may be called geometric codes
in that the incidence vectors of certain flat spaces in the finite
projective geometry PG(t,q) or the finite affine geometry EG(t,q) are
vectors of the dual code.
Such geometric codes are related to
incidence matrices of points and d-flats in the corresponding geometry.
An upper bound is given for the rank of these incidence matrices over
GF(q).
Exact results are obtained for the case d
case d < t-l and prime q.
= t-l
Some known geometric cyclic
and for the
code~
are
reviewed and several new classes of such codes are constructed.
are given which list the parameters of some of the new codes.
Tables
One-step
and multi-step majority decoding algorithms for geometric codes are
given.
A formula for the number of information symbols in certain
Generalized Reed-Muller codes is derived.
Some results on the weight
distribution of Bose-Chaudhuri-Hocquenghem codes are also obtained.
v
SUMMARY
The theory of error-correcting codes is concerned with the development of techniques by which messages can be transmitted accurately and
efficiently over noisy channels.
of transmitting discrete symbols.
We shall consider a channel capable
A message produced by an informaticn
source is encoded into a sequence of symbols which are successively
transmitted over the channel.
Because of the presence of noise in
the channel, a received symbol may differ from the transmitted symbol.
At the receiver, an attempt is made to recover the transmitted sequence
of symbols, based on the received sequence and the characteristics of
the channel.
This operation is referred to as decoding.
This dissertation will be concerned with block codes, in which a
message corresponds to a block of k symbols.
are transformed into blocks of N
over the channel.
=k
These blocks of k symbols
+ r symbols, which are then sent
That is, to each sequence of k information symbols
are added r redundant symbols to form a codeword of N symbols.
of all codewords constitutes the code.
The set
The receiver receives the
sequence of N symbols, some of which may be in error, and attempts to
determine which codeword was transmitted.
Because the code does not
consist of all possible sequences of N symbols, the occurrence of a
few errors does not necessarily change one codeword into another codeword.
This allows the receiver frequently to recognize that error has
occurred, and to correct a certain number of errors.
We shall assume the channel is capable of transmitting anyone of
q distinct symbols, where q is a prime power, say q = pn.
We put the
q symbols into a one-to-one correspondence with the q elements of the
vi
Galois field GF(q).
In this case, a block code consists of a subset
of N-vectors with elements from GF(q).
It is useful to restrict the
set of codewords to be a vector subspace of the vector space V of all
N
N-vectors with elements from GF(q).
code.
Such a code is called a linear
The orthogonal complement of a linear code C, in the usual
algebraic sense, is also a subspace of V and is referred to as the
N
dual code of C.
the dual code.
Each codeword of C is orthogonal to every vector of
This fact is used in decoding a linear code.
This dissertation deals primarily with a decoding procedure for
certain linear codes called majority decoding, as introduced by Massey
[18].1
Majority decoding of a linear code is based on a certain
"orthogonality structure" the dual code may possess.
This orthogonality
structure is related to the incidence relationship in an incomplete
block design.
We may develop this in two directions.
First, suppose
N is the bxv (0,1) incidence matrix of, say, a balanced incomplete
block design with v treatments and b blocks, each treatment being
replicated r times and any pair of treatments occurring together in
A blocks.
The rows of N generate a vector subspace of the vector space
V of all v-vectors with elements from GF(q).
v
The dimension of this
subspace is equal to the rank of N over GF(q).
We may consider this
subspace as the dual code of a linear code C of 1e ngth v and redundancy
equal to the rank of N over GF(q).
In Chapter III we will describe a
majority decoding procedure which may be applied to the code C.
The
main problem in this approach is the determination of the rank of N
over GF(q),i.e., the redundancy of the code C.
The numbers in square brackets refer to references listed in the
bibliography at the end.
1
rii
On the other hand, we may consider a given code and determine
whether the dual code contains a set of vectors which may be considered
as the incidence vectors of a particular incomplete block design, in
which case a majority decoding algorithm will be applicable.
In this dissertation we shall consider majority decoding of linear
codes whose duals either are derived from incomplete block designs
based on certain geometric configurations in a finite geometry over
the field GF{q) or are composed of vectors which are identifiable with
the incidence vectors of such a design.
Chapter I serves as an introduction, in which the coding problem
for a discrete channel is described.
The concept of majority decoding
of linear codes is introduced in this chapter through an example.
The purpose of Chapter II is to investigate the rank, over GF{q),
of the incidence matrix of an incomplete block design obtained by
considering as treatments the points of the finite projective geometry
PG{t,q) and as blocks the d-flats in the geometry.
Analogously, we
consider also the corresponding design in the finite affine geometry
EG{t,q).
We first develop the necessary algebraic and geometric con-
cepts needed to consider these designs analytically.
We make use of
a representation of PG{t,q) and EG{t,q) in terms of an extension field
of GF{q).
Corresponding to each d-flat in the geometry, we define an
incidence polynomial with coefficients one or zero.
The rank of the
corresponding incidence matrix depends on the zeros of the incidence
polynomials in the extension field mentioned above.
Through an investi-
gation of the zeros of these incidence polynomials, we obtain an
explicit formula for the rank of the incidence matrix of points and
(t-l)-flats in PG{t,q) and in EG{t,q).
When q is a prime, i.e., when
viii
q = pn , n=l, we obtain a formula for the rank of the incidence matrix
of points and d-flats in PG{t,p) and EG{t,p) for arbitrary d.
In the
n
general case, q = p , n 2: 1, we obtain an upper bound for the rank of
the incidence matrix of points and d-flats in PG(t,q) and in EG(t,q).
In Chapter III we return to the coding problem and discuss cyclic
codes and majority decoding in more detail.
In particular, we
generalize Massey's concept of majority decoding and investigate the
applicability of majority decoding to some known classes of cyclic
codes.
Using the results of Chapter II, we give a geometric inter-
pretation of the Generalized Reed-MUller codes [15] and obtain an
explicit formula for the number of information symbols for these codes.
We define a new class of codes, which we call Non-Primitive Generalized
Reed-MUller codes, and give both a geometric interpretation of the dual
codes and a formula for the number of information symbols for the codes.
Applying the geometric interpretation of the dual code, we show that
both the Generalized Reed-Muller and the Non-Primitive Generalized
Reed-Muller codes may be decoded by a majority decoding procedure.
Two other classes of codes are defined in Chapter III.
We define
a class of Projective Geometry codes and show that these codes are the
codes considered by Rudolph [21] and investigated recently by Goethals
and Delsarte [10].
A lower bound for the number of information symbols
and a majority decoding algorithm are given for these codes.
Analo-
gously, we define a new class of Affine Geometry codes and obtain
similar results for this class of codes.
CHAPTER I
Ilf.rRODUCTION
1.1 The coding problem
The theory of error-correcting codes is concerned with the
development of techniques by which messages can be transmitted
accurately and efficiently over noisy channels.
We shall consider a
channel which is capable of transmitting any one of q distinct symbols.
Such a channel is called a q-ary channel.
For q a prime or a prime
n
power, say q=p , we may, and do, put the symbols into a one-to-one
correspondence with the elements of the finite field GF(q).
Given a
n
set of m ~ q messages, we set up a one-to-one correspondence between
the messages and a set C of m distinct N-vectors with elements from
GF(q).
C is called a code; each N-vector of C is called a code vector.
The process of constructing the code C and the correspondence between
code vectors and the messages is referred to as encoding.
To transmit a message over the channel, the N symbols of the code
vector (cO'cl, ••• ,c _ ) corresponding to the given message are
Nl
successively transmitted over the channel.
Due to the presence of
random noise in the channel, a transmitted symbol may be .received as
one of the other q-l symbols.
occurred in transmission.
In this case, we say that an error has
The received vector (rO,rl, ••• ,r _ ) may
Nl
differ from the transmitted code vector, and need not necessarily
belong to C.
At the receiver, a decision is made, based on the
2
information in the received vector, which specifies a unique vector of
C, from which the corresponding message is interpolated.
The process
of specifying a code vector, based on the received vector, and the
associated message is called decoding.
The decoding is correct if the
code vector specified is the corresponding transmitted code vector.
If the decoding procedure necessarily gives a correct reSUlt, provided
at most t errors have occurred in transmitting the code vector, we say
that the code is capable of correcting up to terrors.
It has been found mathematically convenient to restrict C to be a
vector subspace of the vector space VB of all If-vectors with elements
from aF(q).
Such a code is called a linear code.
C, regarded as a vector space, is k
k
q.
.:s If,
If the dimension of
the number of code vectors is
In essence, k symbols in a code vector may be chosen arbitrarily;
the remaining r=N-k symbols must be chosen such that the vector belongs
to C.
The number k is referred to as the number o:f information symbols
of C and the number r is referred to as the redund:ancy of C.
Essen-
tially, each code vector consists of k information symbols and r
redundant symbols.
The ratio kIN is called the trl!l.llsmi ssion rate of
information.
The general procedure in linear code
construc·~ion
a code which has a relatively high transmission
ra·~e
is to construct
of information
and which is capable of correcting a relatively l&:lt"ge number of errors.
At the same time, it is desirable that the encoding and decoding
procedures be simple and economical to implement.
Recent work in coding theory has concentrated on cyclic codes.
linear code C is cyclic if, for every code vectorl~:'I=(cO,cl, ••• ,cH_l)
1
The notation £ I denotes a row vector; £ denotes a. colwnn vector.
A
3
of C, the vector (cN_l,cO, ••• ,cN_2)' obtained from
£'
by cyclically
permuting the coordinates one unit to the right, is also a code vector
of C.
Cyclic codes will be discussed in Chapter III.
The synnnetry
induced in a cyclic code simplifies encoding and decoding considerably.
This dissertation will deal primarily with a particular type of decoding scheme, called majority decoding, and its applicability to certain
classes of cyclic codes.
We describe a majority decoding briefly in
the next section and in more detail in Chapter III.
1.2 Majority decodipg
If C is a linear code with length N and k information symbols,
we may consider C as a vector subspace of VB with dimension k.
The
orthogonal complement of C, in the usual algebraic sense, is a subspace of V with dimension r=B-k and is denoted here by Cn.
N
called the dual code of C.
C is
n
Every code vector of C is orthogonal to
Let ~'=(hO'~'.'.'hx_l) be a vector of Cn.
every vector of Cn'
For
an arbitrary vector &'=(gO,gl'.'.'~-l) of VB' let
~'& = hogO+~gl+"'+~-lgN-l = s,
say.
We shall refer to this equation as a parity check equation on
the symbols
gO,gl""'~-l'
and only if
~'&
= 0 for
The vector &' is a code vector of C if
every~' €
Cn'
Suppose i'=(tO,tl, ••• ,tN_l ) is a transmitted code vector and
~·=(rO,rl, ••• ,r._l) is the corresponding received vector.
(unknown) error vector is ~'=(eO,el""'~_l)' where r '
For each vector
~I
€
Cn' we have
______
s = h'r = h't + h'e
since
!! 'i = o.
We
may
write this as
= _h'e:::.J
The
=t'
+ e'.
4
h of
By choosing appropriate vectors
CD' say
~j!'
j=l, •.• ,b, we
obtain a system of equations
(1.2.1)
hjOeO+hjlel+ ••• + hj,N_lelf_l = s,1' j=l, ... ,b.
If the vectors
!!.' j
are known at the receiver, the s j can be determined
for a received vector !.' and, in some cases, the :3ystem of equations
(1.2.1) may be solved for the unknown error symbols eO,el" •• ';'-1'
A relatively simple decoding procedure of this nature, called
majority decoding, is applicable to some linear codes.
Majority
decoding of cyclic codes is discussed in detail in Chapter III.
We
shall illustrate the concept here with an example ..
Consider the following system of equations (over GF(q) ).
+ e
eO + e 1
e
l
+ e
e
(1.2.2)
:: 8
+ e4
2
2
:: s
3
+ e
e
+ e
3
3
e
+ e
l
eO
+ e
::
s4
::
s5
+ e6
::
s6
+ e6
::
s7
+ e6
+ e4 + e
eO
5
5
2
2
= s3
5
+ e4
1
Suppose at most one of the symbols in the vector (eo' e , .'" e6)
l
is non-zero.
Let us restrict our attention to the first, fifth and seventh
equations, viz.,
eO + e l
(1.2.3)
+ e
+ e
eO
e
::
3
+ e
2
4
+ e
5
sl
= s5
+ e6 :: s
7
If each of eo' e , • • " e6 is zero, then each of sl,s5 and s7 is
l
0
5
equal to eo
= O.
If eO is non-zero and each of the remaining symbols is zero,
then again, each of sl' s5 and s7 is equal to eO.
If eO is zero and
e2 , say, is non-zero, then sl and s5 are zero and s7 is non-zero.
In
general, if eO is zero and one other symbol is non-zero, then at most
one of (sl,s5,s7) is different from eO' since eO and any other symbol
occur together in exactly one of these three equations.
In any case,
the majority of (s1' s5' s7) will be equal to eO.
Thus the decoding rule which assigns to eO the value assumed by
the majority of (Sl,s5,s7) will be correct, provided at most one of
the symbols e ,e , ••• ,e6 is non-zero, i.e., provided at most one error
O l
has occured in transmission.
Similarly, by considering the first, second and sixth equations
in (1.2.2), a majority decision rule which assigns to e
1
that value
assumed by the majority of (sl' s2' s6) will be correct, provided at
most one of the symbols e ,e , ••• ,e is non-zero.
6
O l
This may also be
done for each of the remaining symbols e , ••• ,e6.
2
The above procedure is called majority decoding.
It is particu-
larly simple, in this case, and the procedure will correct up to one
error.
The majority decoding is based on the following structure of
equations (1.2.2).
equations.
Each of
~he
7 symbols occurs in 3 of the 7
Any two of these symbols occur together is exactly one of
the 7 equations.
Also each equation contains 3 symbols.
Thus, the
symbols and equations in (1.2.2) form a balanced incomplete block
2
design with parameters
v=7,b=7,r*=3,k*=3 and
~l.
Of particular
Traditionally, r and k are used as parameters of the design. We use
r* and k* to avoid confusion with the redundancy and information
symbols of a code.
2
6
importance here is the property that r*=3 and
~l.
This suggests constructing a linear code of length v with the
property that a set of b parity check equations, which correspond to
vectors in the dual code, exists such that the symbols in a v-vector
and this set of parity check equations form a balanced incomplete
block design with parameters v, b, r*, k*, A, or
gene~rally,
form any
incomplete block design.
Suppose we take such a design of b blocks aD.d v treatments, say,
and consider the b x v incidence matrix, N, defill.ed as
B
= (nij ),
where
1, if the jth treatment is contained in the ith 'block,
0, otherwise; i=1,2, ••• ,b, j:l,2, .•• ,v.
Define a linear code C with length v and with symbols from GF ( q)
such that the row vectors of 5 generate the dual code of C.
Each row
vector of 5, which we call the incidence vector of the corresponding
block, determines a parity check equation for C.
A majority decoding
procedure for C may be used, depending on the typ,e of design and the
parameters of the design.
Of particular interest in this case is the riank of the incidence
matrix 5 over ar(q).
!he redundancy r of the code C is the dimension
of the dual code, which is equal to the rank of 5 over ar(q).
For an
arbitrary incomplete block design, the rank of thle corresponding
incidence matrix over GF(q) is alJoost entirely unknown and appears
to be difficult to treat analytically.
structure of some
desi~
However, "lihe algebraic
derived from finite geometries provides a
7
means of investigating the rank of the incidence matrix of such a
design.
We shall devote most of Chapter II to this study.
The results obtained in Chapter II will be applied in Chapter III
to the construction of some new cyclic codes and to an investigation
of previously known cyclic codes for which a majority decoding
procedure, based on the incidence structure of the dual code, may
be used.
CHAPTER II
INCJJ>ENCE MA!RICES IN FIlfITE GEOMETRIES
2.1 Introduction
In this chapter, we shall investigate the rank of an incidence
matrix of points and d-flats in the finite geometries PG(t,q) and
EG(t,q).
For most combinatorial purposes, such an incidence matrix
'WOuld be considered a real (O,l)-matrix.
For coding theory purposes,
it is appropriate to consider the matrix over the field, GF(q), on
which the geometry is based.
We shall briefly sketch the line of attack taken in determining
the rank of an incidence matrix over GF(q), and then introduce "the
necessary algebraic and geometric concepts.
In particular, a repre-
sentation of (t-l)- flats in PG(t, q) and EG(t, q), ·believed to be new,
is used here.
We shall attempt to keep this chapt,er as self-contained
as possible, so that a familiarity with coding theory is not essential
here.
2.2
~!.2
2!. !!!. incidence matrix
Let us consider an incidence matrix of points and d-flats :in
PG(t, q), say.
Suppose the v points and the b d-fll3.ts of the geometry
are denoted PO' PI'·· .,Pv - l and
La' 11.' .. ·,re-l'
re::;pectively.
An
9
incidence matrix, N, of points and d-flats is defined as
N == «n. '»b
xv ,where for i=O, 1, " ' , b-l and j=O, 1, ••• , v-l,
1, if the point P is incident with the flat
n
=
j
{
ij
0, otherwise.
J.J
~.,
l-
We will regard N as a (O,l)-matrix over the field GF(q).
We also define an incidence polynomial of the flat
~.
J.
as the
polynomial 8 (x), given by
i
v-l
8i (x)
=
L
nijx
j
i=O,l, ••• , b-l.
,
j=O
If the points on the flat ~. are P ,P , ••• ,P , then
J.
e
e
e
2
k
l
e
e
e
l
k
2
8 (X) = x +x +••• +x •
i
Suppose ~O' ~l' ••• , ~v-l
are the v distinct roots of
some extension field, K, of GF(q).
v=
I~ 2
II ••°
l
p
f\
f\2
...
f30v-I f3l v-I
in
Then the matrix V, defined as
1
1
f:0
XV -1=0
·..
·..
·..
~v-l
~v-l
2
,
...
~v-l
v-I:
J
is a non-singular vxv Vandermonde matrix over K, and
••• 8 (f3 _ )
0 v l
8 (f3 _ )
1 v l
NY=
.,. .
...
Now the rank of NY over K is equal to the rank of N over K,
since V is non-singular.
Certainly the ran.'lt of NY is at IOOst equal
to the number of its non-null columns.
It is well known [.3,p.2.32]
that since the entries of N are elements of the subfield GF(q) of K,
10
the rank of N over K is equal to its rank over GF' (q) .
Hence the rank
of N over GF(q) is at most equal to the number of integers u,
o ::;, u
.:s v-1, such that
ei (13)
r 0 for
some i.
EquiValently, the
rank of N is at most equal to v minus tha number of integers m,
o.:s m.:s
v-1, such that
ei (l3m)
= 0, for all i=O,1, ... ,b-1.
We shall later show, through an analytical consideration of the
incidence polynomials of flats in the geometry, that this upper bound
on the rank of N is actually attained.
2.3 Finite fields
Before considering finite geometries, we
re~roduce
here some of
the basic results on finite fields, the proofs of which may be found
in [7, chD(J or [9, ch I], for example.
Throughout, we shall let q be a prime or a prime power, say
q
= pn,
where p is a prime and n is a positive integer.
The non-zero elements of GF(q) are solutions of the equation
(
)
2.3.1
x
q-1
- 1 = o.
If Y is a primitive element of GF (q), i. e., a primtive solution of
equation (2.3.1), then the non-zero elements of GF(q) may be represented by 1=y o, y, ••• , y q-2 •
We may extend the field GF ( q ) to the
extension field GF(qN), whose non-zero elements s.atisfy the equation
(2.3.2)
x
qN 1
- -1 = 0
Let ex be a primitive element of GF(qN).
N)
GF ( q
are represented by
The non-zero elements of
a
qN_2
1=a ,a, ... ,a
Now GF(q) is a subfie1d of GF(l') and the non-zero elements of
GF(qN) which satisfy equation (2.3.1) are precisely the non-zero
0
q-1
elements of GF ( q ) ; these may be represented as l=>'V ,y, ••• ,y
, where
11
N
..J"V(q -1) j(q-l) •
y-....
M:>reover, y is a primitive element of GF (q) •
N
We may regard GF (q ) as an N-dimensional vector space over GF(q)
in the following manner.
Every non-zero element of GF(qN) may be
expressed uniquely as a polynomial in ex, of degree at most N-l,
vuth coefficients from GF(q).
a
u
Suppose, for fixed u, 0
N-l
= a o + ala + ••• +
~_la--
where a O' a , ... , ~-l are elements of C1F(q).
l
aU
o
,cl-l
u
~
N
q -2,
,
The correspondence
~ (ao' al , .•• , ~-l)
~ (0 , 0 ,
••• , 0 )
induces a vector space structure on GF(qN) over GF(q).
elements l,a, •••
~
form a basis for GF(qN).
Indeed, the
We say that the
u
u
'\.
N
elements a l ,ex 2 , ••• ,a of GF(q ) are linearly independent over
GF(q) if, for every choice of elements b , b , .•• , b of GF(q),
k
l
2
the equation
ul
bla
implies b l =b =••• =bk=O.
2
+ b a
u2
~
+. •• + bka
= 0
2
u
u
'\.
Otherwise, we say that a 1,a 2, ••• ,a are
linearly dependent over GF ( q) •
Considering GF(qN) as an N-dimensional vector space over GF(q),
most of the following are standard results in linear algebra and are
proved in [3], for example.
By linear functional on GF(qN), we mean a mapping f: GF(qN)-.GF(q)
such that
f(x l + ~) = f(x l ) + f(x~, f(ax )
l
= af(xl )
N
for any elements Xl' x of GF(q ) and any element a of GF(q). The set
2
of all linear functionals on GF(qN) is an N-dimensional vector space
over GF(q), under the natural operations of addition and scalar
12
multiplication, called t1':e dual space of Gr( qN).
It is well known
that a linear functional is completely determined by the image of any
set of basis elements of GF(qN).
Also, each (non.. zero) linear
functional on GF(qN) is a mapping of GF(qN) onto GF(q).
In particular, the trace [1] of an element v of GF(qN) over GF(q),
denoted T (v) here, is defined as
T (v)
= v+
v
q
N-l
+ ••• + v
q
It may be shown that the trace function, T, is a IlLon-zero linear
functional on GF(q").
Lemma 2.3.1.
We prove the following use:f'ul lemma.
Every non-zero linear functional frclm GF(qN) onto GF(q)
may be expressed in terms of the tra.ce T, in the s:ense that there
= T(J,IX),
exists a non-zero element J.L of GF(qN) such that f(x)
x
€
for all
GF(qN).
Proof.
Let J.L
T(J.L1X)
l
and ~ be non-zero element of GF(qN) and suppose that
= T(J..L2 X)
for all x
T(~X) - T(~X)
for all x
€
N
GF (q ).
€
N
GF(q ). Then
= T(~X
- ~X)
N
= T«~
- ~)x)
But T maps GF (q ) onto GF ( q) .
=0
Hence J.1:L
= J..L2
and
there are thus qN-1 di stinct linear functionals of the form T(J..L. ) .
Now there are N elements in any basis of GF(qN).
Since a linear
functional is completely determined by the image of N basis elements,
there are qN_i distinct non-zero linear functionals from GF(qN) onto
GF(q).
Thus each non-zero linear functional must 'be expressible in
the form stated in the lemma.
The kernel of a linear functional f on QF(qN) is the set of all
elements in (JF(qN) such that f(x)
as a lemma, is well known.
= O.
The following, which we state
13
Lemma 2.3.2.
and f , on GF(qN) have the
2
same kernel if and only if there exists a non-zero element, a, of
Two
linear f'unctionals, f
l
GF(q) such that f 2 = af , Le., f 2 (x) = afl (x) for all x
l
2.4 Finite Erojective geometry PG(t,q)
E
GF(qN ).
We first review some of the basic definitions of points and flats
of PG(t,q), using a coordinate representation [7], [4].
Here,
t ~ 2 and q = pn, where p is a prime and n is a positive integer.
A point, P, of PG(t,q) is represented by an ordered (t+l)-tuple,
The (t+l)-tup~
(xO' xl' ••• , x t ), of elements of GF(q), not all zero,
(YO' Yl' ... , Yt ) represents the same point, P, if and only if there
exists a non-zero element, p, of GF(q) such that y. = pX., i=O,l, .•• ,t.
J.
J.
The number of points of PG(t,q) is v = (qt+l_l)/(q_l).
The set of all points whose coordinates satisfy a system of (t-d)
linear independent homogeneous equations over GF(q),
(2.4.1)
...
a t _d , OXO+
••• +at _d, tXt= 0,
is said to be a d-flat, d=O,l, ••• ,t-l.
is a line, etc.
A G-flat is a point, a I-flat
The number of points on a d-flat is (qd+l_l)/(q_l),
and the number of d-flat. is $(t,d,q), where
t+l
t
.
(qt-d+l_l )
t(t d ) - .(9. -l)(q -1) •••
(2.4.2)
, ,q d+l
d
(q
-l)(q -1) ••. (q-l)
An equivalent representation of points and d-flats in PG(t,q) is
given by Carmichael [7,p345] in terms of an extension field GF(qt+l)
of GF(q).
Let ex be a primitive element of GF(qt+l).
Every non-zero
element of GF(qt+l) may be represented either as a power of ex or as a
14
unique polynomial in a, of degree at mst t, with c:oefficients i.n
t+l
Suppose, for u = O,l, ••• ,q
-2,
)
GF ( q.
The correspondence
o'
aU+-/"x
xl' ••• , x t )
t+l
induces a vector space structure on GF(q
) over GF(q).
With the above correspondence, it is easily verified that a point
P of PG(t, q) is represented by a non-zero element, say a u , of GF ( qt+l ).
Since a
V
u
is a primitive element of GF(q), two elements, a 1 and
represent the same point P if and only if
~E
u
2
mod v.
denote the point represented by aU by (au) or by P.
u
u
'.l
2,
We shall
It will be
understood that the exponent of a is to be reduced rood v to give a
standard representative of the point P.
resented by (aO),
(at), ..• ,
The v points are thus rep-
v l
(a - ) or by PO' P , ..• , P - ' respectivev l
l
ly.
We may re-formulate the definition of a d-flat, given by equations
(2.4.1), as follows.
A d-flat is the set of all points (au) such that
u
fi(a) = 0,
i=l,2, ••• ,t-d,
where f , f , ••• , ft-d. are non-zero independent linear f'unctionals on
l
2
GF(qt+l) • We shall later make use of this representation of d-flats
in PG(t,q) for the case d=t-l.
Alternatively, Carmichael [7] shows that a d-flat may be defined
as the set of points
eO
el
eed
+. •• + ada
),
(aOa
+ ~a
e
eO
el
d
are d+l linearly independent elements of
where a , a , ••• a
.(2.4.4)
t+l
GF(q
) and where &0' a , ••• , ad run independentl~, over the elements
l
15
of GF(q) and are not simultaneously zero. The points
eO
el
ed
(a ), (ex ), ••• , (a ) are called defining points of the flat.
The following theorem is due to Rao [22], [23], (24) •
Theorem 2.4. L
Let the points on a given d-flat be represented by
c
c
c
d+l 1
(a l ), (ex 2 ), ••• , (ex k );
k = q
•
q-l
Then the points
for any integer i, also constitute some d-flat.
Proof.
Let the defining points of the d-flat (2.4.5) be
cl
c d +l
cl+i
(ex ), ••• (ex
). It is sufficient to show that ex
,
are linearly independent over GF(q).
... ,
Suppose they are not.
Then
there exist elements bi-' ... , b d +l of GF«~J, not all zero, such that
cl +
c d +l +1
bla
+ ••• + bd+lex
= O.
.1
cl
c d +l
Dividing through by a , we have that ex , ..• , ex
are dependent,
which is a contradiction.
Theorem 2.4.1 shows that, starting with an initial d-flat whose
points are represented by powers of a primitive element,
Cl,
we may
generate other d-flats by successively adding 1, 2, •.• to each exponent.
Not all d-flats so generated will be distinct.
The smallest
integer, g, such that at the gth stage, the initial d-fiat is reproduced is called the cycle of the initial d-flat [24].
of every d-flat is at IOOst v.
same cycle.
Clearly, the cycle
Moreover, not all d-flats need have the
A study of the cycles of d-flats is given in (24) and (31),
in which it is shown that every (t-l)-flat has full cycle v and that
a necessary and suficient condition that there exist a d-flat in
16
PG(t,q) with cycle less than v is that t+l and d+l have a non-trivial
comm::m factor.
Let us restrict our attention to (t-l)-flats and return to the
linear functional viewpoint, in which a (t-l)-flat is the set of all
points, (au), such that
(2.4.6)
where f is a non-zero linear functional on GF(qt+l).
From Lennm. 2.3.1,
there exists an element, sa:y a -i , of GF ( qt+l ), such that for all
x
€
GF(qt+l)
f(x)
= T(a-i x),
T(x)
=x
where
+x
q
+ ••• + x
qt
•
The (t-l)-flat given by equation (2.4.6) is then the set of all points
(au) such that
(2.4.8)
Let us denote this (t-l)-flat by ~ .• Here, i=o,1, ••• ,qt+l_2 •
J.
.
.,
From Lemma 2.3.2, the linear functionals T(a-J..) and T(a-J..) have
the same kernel if and only if there exists a non-zero element, p, of
. , J ..
V
- J .=-(:tX
GF ( q ) such that a
•
Since a is a primiti'V"e element of GF(q),
this holds if and only if iT == i mod v.
if i
l
== i mod v.
It follows
t~at
Hence, E.,
J.
Ea' 11.' ... ,
= E.J. if and only
1Y-l are the v distinct
(t-l)-flats of PG(t, q), where E.J. is given by equatil::>n (2.4.8).
If the point aU is on the flat E., then T(a-ia u) = O. The point
J.
u+l
(
-(i+l)
u+l = T(a-i. a u ) = O. Therefore,
a
is on the flat 1:i +l , for T ex
ex)
the process of cyclical generation of (t-l)-flats from the initial
(t-l)-flat EO' sa:y, yields the (t-l)-flats El ,
I2"'"
-
Ev _l in order.
17
We observe that the cycle of the (t-l)-flat
v.
of
~O
is therefore equal to
Likewise, we could generate all the (t-l)-flats by choosing any
the~.
J. as an initial flat, obtaining the flats ~.J.+ l""'~.J. +v in
Reducing the subscripts mod v, we see that the flat ~.J. also
order.
has cycle equal to v, for any i
2.5.
= O,l, ••• ,v-l.
Finite affine geometry EG(t,q)
We first describe a coordinate representation of EG(t,q), as
given by Carmichael [7] and Bose [4].
A point of EG(t, q) is represented by one and only one ordered
t
t-tuple, (Xl' x2' ••• , xt ), of elements from GF(q). There are q points
in this geometry. We shall refer to the point represented by the null
vector as the origin.
A d-flat is the set of points whose coordinates satisfy a system
of t-d consistent non-homogeneous independent equations
... + altxt = 0
=0
+ ... +
... ...
~O
+ &llXl +
~O
+ ~lxl
~tXt
at_d'O+at_d,l+··· +at_d,txt= 0
The number of points on a d-flat is qd and the number of d-flats in
EG(t,q) is qt-d.(t_l,d_l,q), where. is given by equation (2.4.2).
Alternatively, we may represent EG(t, q) in terms of an extension
field, GF(qt), of GF(q) [24].
Let a be a primitive element of GF(qt).
A point of EG(t,q) is represented by one and only one element of
t
GF(q).
The point represented by the zero element is the origin,
which we shall denote by P.
co
noted P,
u
O:S u
:s qt -2.
the point it represents.
The point represented by aU will be de-
We shall identify an element of GF (qt) with
18
... , a
e
d
be any set of d
indep.~ndent
elements of
The set of points represented by 0 and the elements
GF{qt).
e
(2.5.2)
aU = ala
l
e
+ a
2a
2
e
+
+ ad a
aU
where
d
and where a , a 2 , •.. , ad are elements of GF{q), not all zero, is a
l
d-flat passing through the origin.
The set of points represented by those elements a
w
c
el
ed
a = a + al a
+
+ ad a ,
w
where
and where a , a , ••• , ad are elements of GF{q) is a d-flat passing
l
2
C
through the point a and belonging to the same parallel bundle as the
flat given by equation
(2.5.2).
The equivalence of the above definitions of a d-flat with that
given by equation
(2.5.1) may be seen through the correspondence
aU +-+ {xl' x2 '
o+-+{o,o,
t-l
+ x2a + ••• + xta
is the unique polynomial representation of aU, u=o,1, ••• ,qt-2 •
where a
u
=~
The number of d-flats passing through the origin is • (t-l,d-l,q),
and the number of d-flats not passing through the origin is equal to
-
(qt-d_ l ) , (t-l,d-l,q).
As in the projective case, we may represent (t-l)-flats of
EG{t,q) in terms of linear functionals from GF{qt) onto GF{q), viz.,
a
(t-~-flat
of EG{t,q) is the set of all points represented by those
elements x of GF{qt) which satisfy the equation
f{x)
= a,
t
where f is a (non-zero) linear functional from GF{q ) onto GF{q) and
a is an element of GF{q).
The equivalence of this definition of a
19
(t-l)-flat with that given by equation (2.5.1) is a consequence of the
correspondence (2.5.4).
Lemma 2.5.1 Let E be given by equation (2.5.5).
Then the kernel of
f, denoted Ker(f), satisfies the following equation.
Proof.
Let X and xl be any elements of E.
o
Then
f(XO-X l ) = f(xO) - f(x l ) = o. Hence the RES of equation (2.5.6) is a
subset of Ker(f). Conversely, if y € Ker(f) and x is any element of E,
then f(x-y) = f(x) - f(y) = a -0 = a.
Therefore, y
Lemma 2.5.2.
€
RHS.
Let
li..
Thus x-y
€
E.
But y
= x - (x-y).
This completes the proof of the lemma.
and ~ be (t-l)-flats, given by f l (x)=al and
f (x)=a2 , respectively. Then 11. = ~ if and only if there exists a
2
non-zero element, p, of GF(q) such that f 2 = pf l and ~ = p~.
Proof. Suppose, for some non-zero element, p, of GF(q), f = pfl and
2
~ = pal.
p .,. 0,
On
Then for all x
€
t
GF(q ), f 2 (x)-a2
= p[fl(x)-al ]. Since
f (x)=a if and only if f l (x)=al , Le., ~ = 1: •
2
2
2
E.I. = ~.
the other hand, suppose
From Lemma 2.5.1,
Ker(f l ) = (xO-xl • xO'~
€
E.I.)
= (xO-xl •• xO,xl
€
~)
·
= Ker(f2 )·
Then, from Lemma 2.3.2, there p-xists a non-zero element, p, of GF(q)
such that f 2 = pf l • Let x €~. Then f 2 (x)=a2 • But 11. =~.
fore, f l (x)=al • Hence ~ = pal and the lemma is proved.
Let E be a (t-l)-flat in EG(t,q) given by equation (2.5.5).
There-
From
Lemma 2.3.1, there exists a non-zero element of GF(qt), saya-i, such
that f(x) = T (a-ix) for all x
l
€
GF(qt), where
20
Tl(x)
=x
+ x
q
t-l
+ ••• + x
q
T is the trace from GF(qt) to GF(q). We use the subscript to disl
tinguish it from the trace from GF(qt+l), us,ed in the projective case •
.,
Suppose~' is the (t-l)-flat given by the eq~ation Tl(a-~ x) = a'.
From Lerrnna 2.5.2,
~
= t' if and only if ther'e exists a non-zero
element, p, of GF(q) such that a
-i'
= pa
therefore, without loss of generality,
flat,
~,
-i
and a' = pa.
choos.~
We may,
the equation of a (t-l)-
to be
where either a=O or a=l.
Since the trace of 0 is 0, the flat
~
passes
through the origin if and only if a=O.
Let us denote the (t-l) -flat through thE~ origin, given by
T (a-ix) = 0,
l
by t~, i=O,l, ... ,qt_2 •
~
Since a
V
'
is a primitive element of GF(q),
where v' = (qt_l)/(q_l), if follows from Lemma 2.5.2 that
and only if i
'5
i, mod v'.
The flats
I1., ..., t;, -1
000
~o'
IP.
~
= t~ if
~
are the v'
distinct (t-l)-flats through the origin in EG(t,q).
In an anaJo gous manner, let us denote the (t-l)-flat not passing
through the origin, given by
T (a-ix) = 1,
l
by t~.
~
Since the RHS of this e:quation is taken as 1 for every (t-l)-
flat not passing through the origin, it follows from Lemma 2.5.2 that,
for i,i' = 0,1, ••• ,qt_2 , the flat ~~ = t~, if' and only if i' =i •
~
~
The
t
111
flats to' E1 , ••• , ~qt -2 are the distinct q -,1 (t-l)-flats not passing
through the origin in EG(t,q).
21
2.6 Incidence matrices
we
2£ points
~
(t-l)-flats
shall treat the projective case first.
The representation of
t+l
points of PG(t,q) as elements of the extension field, GF(q
), of
GF(q), considered as a (t+l)-dimensional vector space over GF(q); and
the representation of (t-l)-flats of PG(t,q) in terms of the kernel of
a linear functional, expressed by the trace from GF{qt+l) to GF{q),
enable us to determine the rank of the incidence matrix of points and
(t-l)-flats in P.F{t,q).
Recall that the {t-l)-flat ~.J. is the set of points, (au), such
that T(a -ia u)
= 0,
. given by equation
i=O,l, ••• ,v-l, where T J.S
(2.4.7)
and a is a primitive element of GF{qt+l).
If we define, for i=O,l, ••• ,v-l, the function
(2.6.1)
then
Ii(etJ)
=Jl,
if the point (etJ) is incident with the flat
to,
Let
nij
~i'
otherwise, j=O,l, ••• ,v-l.
= Ii{etJ),
for i=O,l, ••• ,v-l and j=O,l, ••• ,v-l.
The
matriX, N, defined as
N =
«n. j » vxv
J.
is an incidence matrix of points and (t-l)-flats of PG(t,q), for
n J.J
..
= {l, if the point (~) is incident with the flat
0, otherwise.
M::>reover, considering subscripts mod v,
ni+l,j+l
(2.6.2)
= n .. '
J.J
1:.,
J.
22
Hence, N is a circulant matrix.
We shall regard N as a (0, I)-matrix
over GF(q).
The incidence polynomial of the flat 1:i is defined as
v-I
j
(2.6.3)
B. (x) = \' n ..x •
L
1
From equation
(2.6.2),
lJ
j=O
it follows that
XBi(x) e Bi+l(x), mod xV_I,
and hence, for any integer h,
(2.6.4)
the subscripts taken mod v.
Let 13 = a q- l •
13 is a primitive root of xV_l=O in GF(qt+l) and
the matrix, V, defined by
1
1
1
v =
,
1
13v-I
(132 )v-I •••
(~V-l)v-I.
I
is a non-singular Vandermonde matrix over GF(qt+l) of orde~ v.
From equations
(2.6.3)
NV
= V Q,
and
(2.6.4),
it may be verified that
where Q is a vxv diagonal matrjx given by
Q
=
\--
6 (13
o
V
-
l)
,
Since V is non-singular over GF(qt+l), the rank of N over GF(qt+l)
23
is equal to the rank of Q over GF(qt+l), which is equal to the number
of non-zero elements on its main diagonal.
But, as mentioned in
Section 2.2, the rank of N over GF(qt+l) is equal to its raru~ over
GF(q), since the entries of N are all elements of GF(q).
Hence, we
have the following theorem.
Theorem 2.6.1.
The rank, over GF(q), of the incidence matrix, N, of
points and (t-l)-flats in PG(t,q) is equal to the number of integers,
q l
m, 0 < m < v-l, such that 6 (~m) f 0, where ~ = a - , a is a primitive
0
t+l
element of GF(q
),
v-l
q-l} j
6 (X) =
x ,
{l- [T(~)]
L
0
j=O
and
T(x) = x + x
q
+
...
+ x
t
q
•
We are thus led to investigate the roots of 6 (x) am:mg the roots
0
of xV_l in GF(qt+l).
We may dispense with unity inun.ediately, for 9 (1)
0
is equal to the number of points on a (t-l)-flat, which is
(qt_l)/(q_l).
Thus, 6 (1)
0
In GF(qt+l), an integer is taken mod p, where q=pn.
f
0, for (qt_l)/(q_l) = 1 + q + ••• + qt-l
=1 mod p.
We shall now restrict m such that 1 < m < v-l.
The following series of lemmas will lead to the number of integers
m such that 6 (~m)
0
Lemma 2.6.1.
(2.6.6)
f
O.
Define the function G(x) by
qt+l_2
G(x) =
I
{l -
[T(~)]q-l}xj.
j=O
t+l
Then, over GF (q
),
6 (X)=
0
v
-G(X) , mod x _1.
24
Since a vis a primitive element of G1~ ( q), for any integer h,
Proof.
[T(~+hV) Jq-l
q
t+l
Then
-2
I
=
la
T(~) Jq-l
hV
q-2 v-I
{I - [T(~)]
q-l}j =
j=O
I I{
1 _
h=O j=O
q-2 v-I
= I I {l-
=
[T(~) Jq-l.
[T(~+hV)]q-l}xj+hV
[T(~)]q-l}xj+hV
h=O j=O
v-I
= (q-l)
I
j
{I - [T(CXj)]q-l}x , mod xV_l
j=O
Since q-l
E -
1 mod p, the lemma follows.
We may write 1 - [T(x)]q-l as a polynomi.al in x of degree at most
qt+l - 2
WJ.'th
' UI.'
~(t+l)
coeff"~c~ent s J.n
q
•
Suppos:e
t q-l
1 - [T(X)]q-l =
(2.6.7)
1 - (x+xq+ ••• +x
q )
t+l - 2
q
I
=
u
a x ,
u
u=O
where the coefficients a
Lemma 2.6.2.
u
are yet to be determined.
For 1 ~ u ~ q
t+l
-2,
G(a
-u
)
= -au'
where au is given by
equation (2.6.7).
Proof.
From equation (2.6.7),
qt+l_
2
t+l_ 2
I (r
= I
I
G( -u) =
i=O
j=O
q
t+l 2
-
q
a
i=O
Since
i j } -uj
a,aa.
J.
t+l
i
-2
a(i-U)j
j=O
ex is a primitive element of aF(qt+l ),
i t is well known (see
Van der Waerden [27,p.115], for example) that
25
q
t+1
I
-2
a(i-u)j
j=O
Now for 1
.:s
only if i
= u.
u
.:s
t+l
j l , if i-u a 0 mod q
-1,
=
L 0,
otherwise,
t+l
t+l
t+l
q
-2 and 1 < i .:s q
-2, i-u a 0 mod q
-1 if and
= -au •
Hence
We shall need the concept of the p-adic representation of an
intege~
details of which may be found in any elementary number theory
text, such as [16,PP.9-12].
Every positive integer
may be uniquely expressed as a formal
u
power series in p, of the form
u
where
= Co
+ c l P + c2 P
2
0 < c. < p-l, i=o,l, ••• ,k.
-
1
The coefficients c, are called the
-
digits of u to the pase p.
+ •••
1
We shall denote the sum of the digits of u
to the base p by D (u), i.e., D (u)
P
p
= c 0 +cl+••. +c k '
Let us now return to the expansion of
nomial in x.
1 - [T(x)]q-l as a poly-
Since the fields in which we are 'WOrking have character-
istic p, we may write
+ x
(xP
(2.6.8)
recalling that q
=p
j
+
xP
q
q-l
t
)
j+n
+ ••• +
x?
j+tn p-l
)
,
•
Using the multinomial theorem, the jth term in the above product is
j+tn p-l
j
j+n
(~ + xP
+
+ xP
)
...
p-l
)
j
c.p +c,+ p
x J
J n
j+n
+••• +c,+t p
J n
c j +n '···' c j +tn
where the summation is taken over all choices of c j ' c j +n "··' c j +tn
j+tn
,
26
such that
(2.6.10) c. + c
J
j +n
+ ••. +
t
C.
J+ n
= p-l; 0 < c j +. < p-l, i=O,l, .•. ,t.
~n -
Dickson [8] shows that none of the nultinomial coefficients of
the form
,
ao+~+ •••
1.
where
+a =p-l, 0 < a. < p-l, i=O,.l, ••• ,s, are divisible by p.
S
-
~-
Substituting equation (2.6.9) into equation (2.6.8), we have
t
(x + x q + .•• + x q ) q-l
(2.6.11)
C
~ I ..·ICo,cn,.~~~cJ
.
n- 1,:-11+'
n- n ••• ,cn- l+t)
n
where each summation is given as in equation (2.6.9).
None of the
multinomial coefficients in equation (2.6.11) are congruent to zero
MJrover, each exponent of x is the unique p-adic representat+l
t+l
tion of an integer u, in the range 1 < u ~ q
-1, in fact, u ~ q
-2,
rod p.
for t
> 2.
Let St be the set of all integers u, 1 ~ u ~ qt+l_ 2 , such that if
n-l+tn
c +c 1P+••• +c 1 t p
is the p-adic representation of u, the digits
o
n- + n
of u satisfy equation (2.6.10), for each j=O,l, ••• ,n-l. We observe
n-l
that if u € St' u
(p-l) + (p-l)p + .•• + (p-l)p
=q-l, mod q-l,
=
from equation (2.6.10), i.e., u
1 _ [T(x)] q-l
=0 mod q-l.
=1
u
-
I
€
St
u
b x ,
u
27
and the following lemma is immediate.
t+1
2
Lennna 2.6.3
u
1 - [T(X)]q-l :
a x ,
u
u=O
I-
(i)
where
a
(ii)
o = 1;
if u
(iii)
€
St' au = -bu
otherwise, a
u
=
f
0, over GF(q
t+l
);
o.
We are now in a position to return to a consideration of 6 (x).
0
For mathematical convenience, we shall consider 60(~-m), for
1
From Lemma 2.6.1, 60{~-m) = -G{~-m).
< m < v-1.
G(~ -m) = G(-u)
a
= -au' from Lemma 2. 6 .2.
Setting u = m(q-l)
Hence, 60 ( ~ -m) = au.
Applying
Lemma 2.6.3, we have
Theorem 2.6.2
m(q-1)
€
6 (1)
0
f
0; for 1 ~ m ~ v-l, 80{~-m)
f
0 if and only if
St'
We count the number of elements in St as follows.
Recall that St
is the set of all integers u, such that if the p-adic representation
of u is
u =
Co +
clP + ••• +
then, for each j=O,l, ••• ,n-l,
c j + c j +n + ••• + C j +tn = p-l.
If B(t,p) is the number of ways of choosing t+1 integers dO, •.• ,dt
such that 0
~
d
i
~
p-l,i=O,l, ••• ,t, and d +d1+••• +dt =P-1, then denoting
O
the number of elements in St by 1St I, we have
IStl
Lemma
=
(B(t,p)}n.
2.6.4
B(t,p) = (P+~-l) •
Proof.
p-1
It is well known that B{t,p) is the coefficient of x
(real) expansion of (1 + x + ••• + sP-l)t+1.
Now
in the
28
p-l t+l
(1 + x + ••• + x
)
l' l-~J
=
t+l
I-x
= (l_~)t+l(l_x)-(t+l)
t+l
=
I
g=O
~
(-l)g
[t;~xPgI (t~~xh,
h=O
l
and it is easily verified that the coefficient of xP - in the above
expansion is
(t+i- l )
Thus
.
= (p+~-~
1S t I
n
It was earlier observed that every element in St is a multiple
of q-l.
Combining Theorems 2.6.1 and 2.6.2 with the above, we have
Theorem 2.6.3
1
The rank of the incidence matrix, N, of points and
(t-l)-flats in PG(t,q), where q=pn, is
.
n
p+t-l
1 + \ t
!
\
For the case t=2, this result has been obtained by Graham and
MacWilliams [11], and, for general t, was conjectured by Rudolph [26]
to be true.
Corollary 2.6.1
Li of PG(t,q),
If 9i (X) is the incidence polynomial of the (t-l)-flat
i=O,l, ••• ,v-l, the number of integers m, 0 ~ m ~ v-I,
such that 9.(~m) = 0 for all i=O,l, ••• ,v-l is
J.
. p+t_l\n
v -1.
"
t
/
We shall now consider the affine case.
Because of difficulties
arising in constructing an analytical expression for the incidence
relation between the origin and (t-l)-flats in EG(t,q), we shall
1
This result has been discovered independently b:Y' Goethals and.
Delsarte [10].
29
analyze separately the incidence matrix of points and (t-l)-flats
passing through the origin and the incidence matrix of points and
(t-l)-flats not passing through the origin.
Recall that ~ is the (t-l)-flat, passing through the origin,
which is given by the equation Tl(a-ix)
VI
=
= 0,
i=o,l, ••• ,v'-l, where
(qt_l)/(q_l), and a is a primitive element of G.F(qt).
We observe
that, if I?(x)
is defined as
J.
°
(2.6.12)
=1
Ii(x)
then, for j
-[Tl(a-i x)] q - 1 ,
= o,., .•• ,qt -2,
I~pj) = {I, if the point etJ is incident with the flat
'4.,
0, otherwise.
The incidence polynomial of the flat
nomial
(2.6.13)
ii is defined as the poly-
t
o~(x) =~2
It could be noted here that if t is replaced by t+1, the po:W nomial
9?(X)
is similar to the polynomial G(x), defined in equation
J.
in fact, for i
= 0,
From equation
eg(x)
= G(x).
(2.6.12), we have
x9o()
i x
t
E!
q -1I ,
90i +1 (x) , mod x
and therefore, for any integer h,
(2.6.14)
° ()
t
q -1 -1,
x h 9o()
i x == 9i +h x , mod x
the subscripts being taken mod v'.
xVI
In particular,
t
9.o()
x == e.o()
x , mod xq -1 -l.
J.
1
This implies that
(2.6.15)
l o t
q 1
=
r(x)(x - -1),
1
(xv -l)e.(x)
for some polynomial r(x).
(2.6.6);
30
The origin is not accounted for in the incidence polynomial,
a~(x), of the (t-l)-flat, ~, passing through the ()rigin.
of points on
o
Ii,
other than the origin, is q
The number
t-l.D
-1. Hence, Bi(l) =-1,
over GF(qt).
° :S m =s qt -2,
From equation (2.6. 14), for any integer m,
°
Therefore,
equation
eo(a-m) = (-m)ieO(
a
i a -m) , 1"=0 , 1 , ••• ,v
eO( a -m) =
if and only if eO (a-m ) = 0.
i
°
°
(2 ••
6 15 ) that, unless m
is a primitive root of xv'_l
=
50 mod q-l,
I
-1.
It follows from
"
eOo( a m) = 0, Slnce
a q-l
° in GF(qt).
°
We observed earlier that if t is replaced by t+l, eo(x) = G(X),
where G(x) is defined in equation (2.6.6).
lennnas to investigate eOo( a-m) for m
!i!3
We may thus use the previous
° mod q-l.
lfe summarize these
results in the following theorem.
Theorem 2.6.4
r..g
Let
e~(x) be the incidence polynomill.l of the (t-l)-flat
passing through the origin in EXt ( t , q) •
(i)
e~(l) = -1;
(ii)
Unless m
5!
° rod q-l,
(iii) For 1 =s m =s v'-I,
m( q-l)
€
Then
eg(am) = 0;
e~(a-m(q-l» ~
St_l.
t
There are q -1 -v' =(q-2)v' integers h,
h
'!=
° if and only if
° mod q-l.
t
° =s h =s q -2,
t
such that
Since 1St_II
0=Sm=sq-2 such that
= (P+t-2) n, the number of integers m,
t-l
e~(a-m) ~ is
°
1 + (p+t_2)n.
t-l
By adjoining a column of ones to the incidenCE! matrix, N, of
point~
other than the origin, and (t-l)-flats passing through the origin in
EG(t,q), we may correspond the additional column to the origin and
31
consider the augmented matrix as the incidence matrix of (all) points
and (t-l)-flats passing through the origin in EG(t,q).
' by N* •
augment e d mat r~x
S'~nce there are qt - l
. t s on a (t - 1) - fl a,
t
po~n
this additional column is dependent (over GF(q»
matrix N.
Denote this
on the columns of the
*
Thus the rank of N is equal to the rank of N.
From
Theorem 2.6.4, we have
Theorem 2.6.5
Over GF(q), where q
= pn,
the rank of the incidence
matrices N and N*, defined above, of points and (t-l)-flats passing
through the origin in EG( t, q) is equal to 1 +
(P~:~2 ') n
•
ti is a (t-~flat in EG(t,q) which does not pass
Then zi is given
Suppose now that
through the origin.
by
Tl(ciix)
for i=o,1, ••• ,qt-2 •
Let
Iil( x )
(2.6.16)
= 1,
1 - [1 - Tl (-i
a x)] q-l •
=
Then, for j=o,1, ••• ,qt_2 ,
:ri(a1)
{1. , if' the point
=
~
is incident with
C, otherwise.
The incidence polynomial, e:-(x),
of the flat
~
r
l
is given by
t_ 2
B~(X)
=
~ Ii(~)~.
j=O
If the k
=q
t-1
= k
u1
'\
u
, a 2 , ... , a , then
~
are a
1
9 (x)
ul
u2
'\
+ x
+. •• + x ,
=
1
'+1
I i +l (etJ ), we have
=x
i
and 9~(1)
l
1
points on
== 0, mod p.
1
'
Again, since I.l (etJ)
1
x 9,l (x) ==
ei+1 (x),
t
mod sq -1_1,
~,
32
and consequently, for any integer h,
~
qt_ 1
6i+h(x), mod x
-1,
h _1
x 8i(x) ==
(2.6.17)
the subscripts being taken mod qt -1From equation (2.6.17), it follows that, for 0 ~ m ~ qt_2,
ei(cfl)
= 0, for all i=0,1, ••• ,qt-2, if and only if
'o(cfl)
= 0.
~(a-h), where 1 ~ h ~ qt -2.
It will be convenient to consider
t
q -2
~(a-h)
{I - [1 - T (a1) ]q-l}
1
= I
a -jh
j=o
qt_ 2
(2.6.18)
=
I
qt_ 2
[1 - T (a1) ]q-l a- jh •
l
a -jh - I
j=O
j=O
t
Now
q -2
L\ '
a-jh
= {-1,
j=O
if
h
==
° mod
q
t
-1,
0, otherwise.
Let us consider the second term in equation (2.6.18).
Recalling that q = pn, we may expand [1 - T (X)]q-l as
1
[1 - T (x) ]q-l
1
=
[1 - (x + x
n-l {
= n
1 j=O
(xP
j
q
+ ••• + x
+ ~
j+n
qt-l
q-l
)]
+ ••• + ~
j+(t-l)n }P-l
)
For j=O,l, ••• ,n-l, each term under the product in eCluation (2.6.19)
is expanded as
{ 1 - (~
(2.6.20)
=
PI-I
u=O
(-1)
u
j
+ ~
j+n
(P-I) [j
xP
u
+ ••• + ~
+ ~
j+n
j+(t-l)n} 1
) p-
+ ••• + x
pj+(t-l)n
-u
J
33
We note that none of the binomial coefficients in the above are
divisible by p.
Now, for 0
"
l
~
j
+~
j+n
.:s u .:s p-l,
+ ••• +~
j+(t-l)n J- u
(2.6.21)
+•.• + cj+(t_l)nPj+( t-l)n
where the summation is taken over all choices of the cts such that
c.J + cj+n +••• + c'+(t
< Jc.+
<u, r=O,l, ••• ,t-l.
J
-1)
n= u, 0 rn-
(2.6.22)
Since 0
.:s u .:s p-l,
none of the multinomial coefficients in equation
(2.6.21) are divisible by p.
SUbstituting all of the above into
equation (2.6.19), we may write
(2.6.23)
=
[1 - T (x)]q-l
l
I
iif€R
w
b x ,
w
t
where R is the set of all integers, w, such that if the p-adic ret
presentation of w is
w
= Co
+ c 1P + •••
+
n-l+(t-l)n
,
n-1 + (t-1)nP
C
then for each j=O,l, ••• ,n-l,
o
.:s
c j + c j +n + ••• + Cj +(t_1)n
We note that if w
t
w .:::: q -2.
€
R , then 0 .::::
t
The coefficient b
w
w.:::: qt_l.
.:s p-l.
In fact, for t
I
j=O
2,
is not congruent to zero, mod p for
Returning to equation (2.6.18), we have, for 1
t
q -2
2:
.:s h .:s
t
q -2,
,
34
t
q -2
Now
I
a(w-h)j =
{-I, if w-he 0 mod qt_l ,
j=O
Since, for 1
= w,
if h
0, otherwise.
.5
I
h
t
q -2 and for WERt' w-h
.5
5:
° mod q -1 if and only
t
t_ 2
j
[1 - T1 (al)]'l-la - h = {-hh' if' h • Rt
0, otherwi ,se.
Equation (2.6.18) becomes, for 1
.5
h
.5
t
q -2,
bh (/=0, modp), ifh €R ,
t
{
0, otherwise.
We collect these results in the following theorem.
Theorem 2.6.6
Let
~(x)
be the incidence polynomial of the flat
not pass ing through the origin in EG ( t, q), i=O, 1, ••., , qt -2.
~
Then
1
e.(l)
= 0;
1
(i)
(ii)
If h
(iii)
If h
>
l( a -h )
1 and h € R ,
2:
9i
t
1 and h € R ,
t
f
0, h
.5
t
q -2;
~(a-h) = 0, h .5 qt -2.
We count the number of elements in R as follows.
t
If C(t,p) is
the number of ways of choosing t integers, d , d , .... , d , such that
t
l
2
o .5
d
i
.5
p-I, i=1,2, ... ,t, and
IRt I =
° .5 d l
(C ( t , p)
f .
+ d
2
+ •.• + d
t
t
.5
p-I, then
35
+ d + •.• + d = p-l - e, where 0 ~ e ~ p-l, we have
2
t
1
e + d + ••• + d = p-1. Hence, C(t,p) = B(t,p), where B(t,p) is
t
1
Now if d
given by Lemma 2.6.4.
Thus,
n
IR t I = (P+~-l) •
Since
0 € R
t , the number of integers h, 1
~
h
~
t
q -2 such that
6~(a-h) f 0 is, from Theorem 2.6.5, equal to IRt
I -1,
which is
n
(P+~-l) _ 1 •
If we define the matrix VI as
'1
VI
=
I
L
1
1
a
1
2
a
1
1
a
ri
(ri)2
t
(aq -2)2
t
1
a
q -2
1
q
t
t
(ri)q -2 ... ( a q
-2
t
,
t
-2)q -2
.J
t
it is easy to verify that VI is a non-singular matrix of order q -1
over GF(qt), since a is a primitive element of GF(qt).
Let us define the incidence matrix of points (other than the
origin) and (t-l)-flats not passing through the origin in EG(t,q) as
the matrix, N, given by
N
= «I~ ( ; ) ) ) •
It follows from equation (2.6.17) that
where Q is a diagonal matrix with
l
.
entr~es
61()
1 , ••• ,61( a qt_ 2 ) •
0
0
As has been argued earlier, it is seen that the rank of N over
GF(q) is equal to the number of integers h, 0
6;(eP) f
~
h
~
t
q -2, such that
o. From Theorem 2.6.6, this is equal to (p+~-~
n
-1,
By adding a column of zeros to N, corresponding to the origin,
36
the augmented matrix N* may be viewed as the incidence matrix of (all)
points and (t-l)-flats not passing through the origin in EG(t,q).
Obviously, the rank of N* is equal to the rank of N.
Theorem 2.6.7.
Hence we have
The ranks of the incidence matrices N and N*
over GF(q) are both equal to
(p+~-]~f
- 1, where N and N* are defined
above.
2.7
Incidence matrices
2! points
and d-flats in
!~(t,q).
We turn to the problem of determining the rank of the incidence
matrix involving points and d-flats, where 1 < d <: t-l.
The projective
case will be treated first.
We shall use an extension of the methods used by Graham and
MacWilliams [11] in our investigation of the rank of the incidence
matrix of points and d-flats in PG(t,q).
In [11], only the case t
=2
was considered.
Recall from Section 2.4 that a d-flat in PG(t,q) is the set of
points
eO
el
ed
(aOa + ala +
+ ada ),
e
e
e
where the defining points aO,a 1, ••• , a d are linearly independent
(2.7.1)
elements of GF(qt+l) and a is a primitive element of GF(qt+l), and
where aO' aI' ••• , ad run independently over the elements of GF(q),
not all zero.
~
~
~
t~
Suppose (a ), (a ), ••• , (a ) are the k = (q
-l)/(q-l)
points on the d-flat 1:, given by equation (2.7.1).
exponents al, ••• ,a.1{ have been reduced mod v, i.e.,
i=l, ••• ,k.
We assume the
0
< a.l < v-I,
-
The incidence polynomial of the d-flat 1: is defined as
the polynomial
37
+x~
e(x)
From Theorem 2.4.1, the polynomial
a +i
xi e(x) = x 1
+
a +i
+ x
k
,
i=O, 1, ••• ,
when reduced mod xV_I, also is the incidence polynomial of some d-flat.
If g is the cycle of the d-flat 1:, then
xge(x)
e(x), mod xV_l.
5
It is proved in [24] that all d-flats in PG(t,q) may be generated
cyclically from a set of initial d-flats which may be of different
cycles.
where 1
I1, I2' ...
Suppose
~ ~ ~
,1:~
is such a set of initial d-flats,
b/v, where b is the number of d-flats in PG(t,q) and
is equal to .(t,d,q).
Let the corresponding incidence polynomial
of 1:.~ be e.~ (x), i=l, ••• , ~.
We say that a given d-flat 1: be longs to
the cycle of the flat 1:. if 1: can be obtained from 1:. by the process
~
~
of cyclical generation, in which case the polynomial of 1:, say e(x),
satisfies xhe.(x)= 9(x), mod xV_I, for some integer h.
~
cycle of the flat 1:i is gi' i=l, ••• ,~.
Suppose the
We observe that gl + ••. +
g~=b.
We may order the d-flats of PG(t,q) such that the first gl flats
are 1:1 and the gl-l distinct d-flats cyclincally generated from it, the
next g2 flats are
~
and the g2- 1 distinct flats generated from 1: , etc.
2
According to this ordering, we define the incidence matrix of points
and d-flats to be the matrix, N) given by N = «n.~J'»bxv' where
n ..
~J
=
{I,
if the point (~) is incident with the ith d-flat,
0, otherwise,
for i=1,2, ••• ,b and j=O,l, ••• ,v-l.
It will be mathematically convenient to disregard the cycle of
a d-flat and consider the v d-flats cyclically generated by each of
38
I: , ••• ,I: •
Not all of these 1rV d-flats will be distinct, if there
n
exists a d-flat of cycle less than v, but every d-flat will be included
l
in the flats so generated.
If N is the corresponding incidence matrix,
then N*.1S a nvxv matrix.
The matrix N can be obtained from N* by
*
deleting duplicates of rows of N.
It may be verified that, if V is
the Vandermonde matrix defined in equation
equation holds.
(2.7.2)
(2.6.5), the following
,
N* V =
nvxv vxv
V
0
0
0
Dl
0
V
0
0
D2
0
0
0
V
l
I
D1C
J
1CVX '1CV
J
fCVXV
where
1
r'
19. (1)
0
i 1
D.1
=
9i (13)
I
I
I
I
I
L
0
,
e. (13V-I)
J.
-
VXV
for i=l, ••• ,n.
Since both V and the composite matrix of V's on the RES of
t+l
equation (2.7.2) are non-singular matrices over GF(q
), the rank
*
of N is equal to the rank of the second matrix on the RES of equation
(2.7.2).
By elementary row operations, the rank of this matrix is
equal to v minus the number of integers m, 0
9.(f3m)
J.
= 0,
for all i
*
= 1,
~
m ~ v-I, such that
••. ,1(.
*
Because N may be obtained from N by deleting duplicates of rows of N ,
39
*
the rank of N is equal to the rank of N.
Noting that the rank of N
t+l
over GF (q
) is equal to the rank of N over GF (q), we have the following theorem.
Theorem 2.7.1.
Let E , ••• ,E be a set of initial d-flats from which
ff
l
all d-flats of PG(t,q) may be Generated, and let a.(x), i=l, ••• ,ff,
~
be the corresponding incidence polynomials.
Then the rank of the
incidence matrix, N, of points and d-flats in PG(t,q) is equal to
v minus the number of integers m, 0 ~ m ~ v-l, such that ai(~m) =
° for
each i=l, ••• , ff.
Suppose E is a d-flat generated from E., for some i=l, ••• ,ff.
~
Then,
denoting the polynomial of E by aE(x), we have
aE(x) e xha (x), mod
i
Hence, aE(~m) =
° if and only if
XV -1,
ei(~m)
for some integer h.
for any m.
= 0,
We therefore
have the following corollary to Theorem 2.7.1.
Corollary 2.7.1 The rank of N over GF(q) is equal to v minus the number
of integers m, 0 ~ m ~ v-l, such that aE(~m)
= ° for
every d-flat E in
PG(t,q).
e
e
e
Let E be a d-flat with defining points (a 0), (a l), ••• ,(a d),
eo
el
ed
t+l
where a , a , ••• , a
are linearly independent elements of GF(q
).
Define the polynomial SE(x) by
where the summation is taken over all integers u such that
for some aO, ••• ,ad of GF(q), not all zero.
Syppose, for a particular
40
eo
el
ed
c+jv
.
aOa +ala +•.• + ada
=a
,0 .:5 c ,~v-l, 0 .:5 J .:5 q-2.
V
c
Then (a ) is a point of~. Since a is a primitive element of GF(q),
(2.7.5)
the element s a C , a C +v , ••• , a C +( q-2)v, which represent the same point
of~,
are also given by equation (2.7.5) for some choice of aO, ••• ,a •
d
Thus, each of the q-l representations of a point of
we
equation (2.7.4).
s~(x)
~
are given by
may therefore write
=
9~(x)
E
(q-l) 9~(X), mod xV_l
E
-
+
9~(x),
XV9~(X)
+ ••• +
x(q-2)v9~(X)
v
t+l
mod x -1, over GF(q
).
We state this as a lemma.
Lemma 2.7.1.
Let ~ be a d-flat with incidence polynomial eZ(x).
S~(x) defined by equation (2.7.3), then e~(x)
E
For
-S~(x), mod xV_l, over
GF(qt+l).
The number of points on a d-flat is (qd+l_l)/(Cl_l) = 1 + q +.•. +qd •
Hence, eZ(l)
=1, mod p,
for any d-flat ~.
Let us consider 8~(~m), where Z is any d-flat lmd 1 < m < v-l.
Since f3m is a root of xV_l=O, we have e~(~m) :: _S~Wm), from Lemma 2.7.L
Now,
S (~m) :: S (am(q-l»
~
~
=
I
(au)m(q-l),
u
where the summation is taken over all choices of u such that
eO
el
ed
aU = aoa + ala +... + ada
for some aO, ••• ,ad of GF(q), not all zero, and where a
defining points of the flat E.
eo
ed
, ••• ,a are
For m > 0, om(q-l)= O. So we may write,
41
for 1 .::: m .::: v-I,
\'
m
S1:(f3 )
\'
eO
ed
(aoa + ••• + ada )
L'" L
=
m(q-l)
,
ad
aO
where each summation is taken over all elements of GF(q).
Expanding
each term, we have
=
L(.
m(q-l)
J O'
j
jl' "',
where the summation is taken over all choices of jO,jl, ••• ,jd such that
jo +jl + ••• + jd
= m(q-l)
and
° .: : jv'::: m(q-l),
V=O,l, ••• ,d.
SUbstituting into equation (2.7.6), and interchanging the order of
summation, we obtain
Lemma 2.7.2
For any integer j
~
a
aeGF q)
Proof.
j
=
{-l, if
~
0,
= k(q-l),
j
k
> 0,
0, otherwise.
It is well known that for any integer j,
= {
-1, if j
=0, mod q-l
0, otherwise
If j
= 0,
then a
result is true.
O
• 1 for all a
€
GF(q) and since q
If j > 0, then since oj
= 0,
= 0,
mod P, the
the result follows from
42
the above.
It follows from Lemma 2.7.2 that, unless each j
v
in equation
(2.7.6) is equal to k (q-l), k >0, for v=O,l, ••• ,d, the corresponding
v
term vanishes.
(2.7.7)
SI:(~m)
v
Thus,
m(q-l)
==
(_l)d+l
I~o(q-l), ••• ,kd(q-l») ex
eoko(q-l)+···+edkd
k
where the summation is taken over all choices of ko(q-l), ••• ,kd(q-l),
such that
d
o < k (q-l) < m(q-l), v=O,l, ••. ,d, and
v
-
I
k v(q-l) == m(q-l).
v=O
Now, a sufficient condition that SI:(~m)
==
0 for every d-flat I:
is that each of the multinomial coefficients in equation (2.7.7) is
congruent to zero, mod p.
We shall show this condition is also
necessary.
Ad-flat I: is determined by a set of (d+l) linearly independent
e
eO
d
...
,
ex
• We may regard SI:(~m) as a j[>olynomial in
elements ex ,
e
eO
e
d
ex , ex l , ••• , ex ; in fact, we see from equations (:2.7.7) and (2.7.8)
that this is a homogeneous polynomial of degree m(q.-l) < qt+l_l.
eO
ed
Suppose ex , ••• , ex were chosen to be linearl:r dependent over
e.
GF{q), instead of linearly independent. Then, one of the ex l,S is a
linear combination of the others.
ex
eO
==
We may assume that
el
ed
b l ex
+ •.• + b d ex without loss of generality, for some
b l , b , ••• , b d of GF{q), not all zero.
2
(2.7.6), we have
SUbstituting this into equation
43
where at i
= aob.+a.,
i=l, ••• ,d.
~
~
Since a
o
does not appear in the
second term of this equation, we have
== 0, mod p
Suppose S~(~m) = 0 for all d-flats~. Then it is zero for all
choices of linearly independent elements a
eO
e
, a l , ... , a
ed
.
It has
just been shown that it is zero for all choices of linearly dependent
elements a
eO
, a
el
, ... , a
ed
also.
e.
Hence, S~(~m) = 0 for all q
t+l
-l
choices of a 1, i=O,l, ••. ,d. Since s~(~m) is a homogeneous polynomial
e.
t+l
in the a l,S, of degree ~ than q
-1, this can happen only when
.
each coefficient
lS
zero, over GF(q
t+l
).
We now investigate under what conditions each of the multinomial
coefficients in equation (2.7.7) is congruent to zero, mod p.
The following lemma is due to Dickson [8,p.273].
Lennna 2.7.3
If u
l
+ ••• + U
s
= u, where 0 < u. < u, i=1,2, ••• ,s, the
- 1-
multinomial coefficient
(U l 'U2 ' ~ .• ,
u)
is prime to p if and only if, when u and each of the u.1 are written
to the base P, in the form
44
u
I
I
n
n
c Pj ,
j
j=O
o -<
c.
J
.:s p-l;
u.
l
=
j=O
c. (i)"i
p"',
J
o -<
c. (i)
J
.:s p-l,
the sum of the jth digits of each u.l is equal to the jth digit of u,
i. e.,
c.
J
(i)
,
j=O,l, ••• ,n.
Moreover,
~
) \ == j =0 ( ()
(u 'u u
, .•. , u
1
l 2
s
Cj
'
c j(2)
Cj
"",c j
(s) "') , mod p.
M:>re insight is obtained as to the divisibility of a multinomial
coefficient through the following lemma.
If Dp (u) denotes the sum of the digits of u to the base p,
then the highest pONer of p dividing the multinomial coefficient
+ D (u ) - D (u) •
p s
p
This lemma is a consequence of the well known result [8,p.263]
that for any positive integer u, the highest power of p dividing u: is
equal to
u - Dp (u)
p-l
It follows from Lemma 2.7.4 that the multinomial coefficient
m{q-l)
"( ko(q-l), ••• , kd{q-l) )
'1= 0, mod p,
i.e., is not divisible by p, if and only if
Dp (m{q-l»
= Dp (kO(q-l»
+ ••• + Dp (kd{q-l»;
45
otherwise,
Dp (m(q-l)) < Dp (kO(q-l)) + .•• + Dp (kp (q-l)).
Let us, for the moment, restrict q such that q = p, i.e., n = 1.
Lemma 2.7.5
Proof.
For any positive integers a and k, Da (k(a-l))ao, mod a-l.
Suppose the a-adic representation of k(a-l) is
N
k(a-l) = b O + bla + ••• + bNa
Then, since a
=1, mod a-l,
.
=bO+bl+••• +bN, mod a-l. But
Since k(a-l) =0, mod a-l, the result
k(a-l)
bO+bl+••• +b = Da(k(a-l)).
N
follow·s.
As a corollary, if k > 0, then D (k(a-l)) = r(a-l), where r > 1.
a
Lemma 2.7.6 For q = p, one of the multinomial coefficients in equation
(2.7.7) is not divisible by p if and only if Dp (m(p-l)) = r(p-l), for
some r, d+l < r < t+l.
Suppose one of the multinom ial coefficients in equation (2.7.7)
Proof.
is not divisible by p.
Then, from Lemma 2.7.4,
Dp (m(p-l)) = Dp (kO(P-l)) + ••• + Dp (kd(p-l)),
where kO, ••• ,k
d
satisfy equation (2.7.8), in particular, k y
•.• ,d.
2 p-l, v=O, ••• ,d. Hence,
From Lemma 2.7.5, Dp(ky{P-l))
Dp(m(p-l))
>o,~O,
2 (d+l)(p-l). For 1 S m S v-l, the p-adic representation of
m(p-l) is
m(p-l) =
Co
Thus, D (m(p-l)) < (t+l)(p-l).
p
-
t
+ cIP + ••• + c t P
We therefore must have Dp(m(p-l))=r(p-lh
where d+l < r < t+l.
Conversely, suppose D (m(p-l)) = r(p-l), for some r, d+l < r < t+l.
p
-
-
We shall show that there exist d+l integers kO(p-l), ••• ,kd(p-l) such
that 0 < ky(P-l)
S m(p-l),
v=O,l, ••• ,d, such that kO(P-l) +••• + kd(p-l)
46
= m(p-l) and Dp (m(p-l»
= Dp (kO(P-l»
+ ••• + Dp (kl(p-l»,
by an
(
induction argument through the following lemma.
For m > 0, let Dp (m(p-l»
Lemma 2.7.7.
for any (j,l <
-
(j
< s, there exist
-
o < k v (p-l) -< m(p-l), v=l, ••• ,
such that Dp (m(p-l»
Proof.
(j
C1',
= Dp (kl(P-l»
= s(p-l), where s > 1.
Then
integers kl(p-l), ... ,k (p-l) with
C1'
and kl(p-l) +.•. + k(j (p-l) = m(p-l)
+ ... + Dp (kC1' (p..l».
Let the p-adic representation of m(p-l) be
t
m(p-l) = Co + c l P + ••• + c P
t
For s = 1, the lemma is trivially true.
For s = 2, let j be the smallest integer such that p-l<c +•.• +c .<2(P-l).
o
JClearly 1 .:::: j .:::: t and O<C + c l + ••. + Cj _ .:::: p-l. Let
O
l
u l = Co + clP + ••. + Cj_1Pj-l + [(p-l) - (CO+cl+· .. +cj_l)]pj
j
j+l
t
u 2 = [ cO+cl+,·,+c j - ( p-l )] P + c j +l P
+ ••• + ctP •
Now
~
and u 2 are both positive and the above are their respective
p-adic representations.
Dp (m(p-l»
u
2
Moreover, Dp(U l ) = p-l and. Dp (U ) =
2
- (p-l) = (s-l)(p-l) = p-L
Thus u
= k (P-l), for some positive k l and k •
2
2
and the lemma is true for s = 2.
l
= k (p-l) and
l
Furthermore, u +u = m(p-l)
l 2
Assume the lemma is true for s'=s-l.
We shall show it is true for s'=s.
Let u
l
and u
2
be as before,
Dp (U2 ) = (s-l)(p-l), and ~ = k2 (p-l), where k > O.
2
Applying the
induction bypothesis to k (p-l), we may write
2
k (p-l)
2
= k'(p-l)
1
+ ••• + k'(p-l), 1 ~ cr.:::: s-l,
C1'
such that 0 < k' (p-l) < k (p-l), v=l, ••• ,
v
- 2
D (kl'(p-l»
p
Letting u
l
+ •.• + D (k'(p-l»
p
C1'
= kO(p-l), the lemma follows.
C1',
Then
and
= DP (k2 (P-l».
Using this lemma, the required set of d+l
~
r integers exists
and, from Lemma 2.7.4, the corresponding multinomial coefficient in
equation (2.7.7) is not divisible by p.
~
Let
be a d-flat in PG(t,p).
This completes the proof of
From Lemma 2.7.1, for any integer
) e~(~m) =-S~(~m), where
m, 1 ~ m ~ v-I, where v = ( l't+l -1 ) / (
1'-1,
~ = c/-1 and a is a primitive element of GF(pt+1).
We thus have the
following theorem.
Theorem 2.7.2
f.
(i) e~(l)
Let
~
be a d-flat in PG(t,p).
For every d-f1at
0;
(ii) For 1 ~ m ~ v-1, e~(~m) = 0, if and only if Dp(m(p-1»
where 1
~
s
~
<s <
= s(p-1),
d, i.e., a necessary and sufficient condition that
e~(~m) = 0 for every d-flat ~ is that Dp (m(p-1»
1
~,
= s(p-1), where
d.
m ~ V-I, we may write the p-adic representation of m(p-1)
t
as m(p-1)= Co + c p + ••• + c t p • The number of integers m, such that
1
For 1
Dp (m(p-1»
~
= s(p-l) is equal to the number of ways of choosing t+1
integers cO',·"c t ' 0 ~ ci
Bs(t,p) be this number.
S 1'-1,
with
Co
+••• + c t = s(p-1).
Let
We observe that B1 (t,p) = B(t,p), defined in
Section 2.6.
Now, Bs(t,p) is the coefficient of s(p-1) in the real expansion
of (1+x+••• +xP- 1 ) t+1
We may write, as in Lemma 2.6.4,
t+1
(l+x+ ••• +~-1) t+1 =
I
g=O
~
(-l)g (t;l)
xP
g
I (t;j
xh •
h=O
If we denote by L (1'), the greatest integer in s(p-1)/p, i.e. ,
s
s(p-1)
s = 1, •.. , t+1,
Ls(p) = [
P
J,
48
it
may
be verified that
Let
d
Rd(t,P)
=
I
Bs(t,P).
s=l
The number of integers m, 1
1 ~
~
m ~ v-I, such that Dp (m(p-l»=S(P-l),
s ~ d, is thus equal to Rd(t, p) • From the corollary to Theorem
2.7.1, we have,
Theorem 2.7.3
The rank of the incidence matrix of points and d-flats
in PG(t,p) is equal to v minus Rd(t,P), where Rd(t,P) is given by
equation (2.7.11).
Now Bs (t,p) is the number of ways of choosing t+l integers
s(p-l) •
Then
° -< d.
< p-l, i=O,l, ••• ,t, and
~-
s =O,l, .•• ,t+l.
Also, Rt(t,P) = Rt _1 (t,p) + Bt(t,P) = Rt _1 (t,:p) + B1 (t,P).
Now
Rt(t,P) is the number of integers m, 1 ~ m ~ v-I, such that
Dp (m(p-l»
= s(p-l), where 1 ~ s ~ t.
has the property that Dp (m(p-l»
Rt(t,P)
But every integer m, 1 ~ m ~ v-I,
= s(p-l), where 1 < s < t.
= v-I. We have, therefore,
R _ (t,p) = v-I - B (t,p),
t 1
1
and from equation (2.7.10),
B1 (t,P) = ~t+P-I)
t
.
Thus
Rt _1 (t,p) = v-I - ( t+P-I)
t
.
Hence
From Theorem 2.7.3, the rank of the incidence matrix of points and
(t-l)-flats in PG{t,p) is equal to v - Rt_l{t,p) = (
t+P-l··
t ) + 1, the
result we obtained in Theorem 2.6.3 for the case n=l.
n
Returning to the general case, in which q = P , n
~
1, we shall
obtain an upper bound on the rank of the incidence matrix of points
and d-flats in PG{t,q).
For 0
S m S v-l,
let the q-adic representation of m{q-l) be
m{q-l) =
where 0
Ac
+ A:Lq + ••• + Atq
t
,
S Ai S q-l, i=o,l, ••• ,t. We have Dq{m{q-l» = Ao+···+At=O,
mod q-l, for m(q-l)=O, mod q-l.
If one of the multinomial coefficients in equation (2.7.7) is
not divisible by p, it is certainly not divisible by q.
Replacing p
by q in Lemma 2.7.4, the latter is true if and only if
Dq (m{q-l»
= Dq (ko(q-l»
+ ••• + Dq (kd(q-l».
Since each k i is positive, Dq(ki(q-l»
~ q-l, i=O,l, ••• ,d.
ThUS, if
one of the multinomial coefficients in equation (2.7.7) is not
di visible
by P, we must have
D (m(q-l»
q
Then, if D (m(q-l»
q
> (d+l)(q-l).
-
< (d+l)(q-l), each of the multinomial coefficients
in equation (2.7.7) must be divisible by q and hence also by p.
We
therefore have the following theorem.
Theorem 2.7.4
(i)
(ii)
eL(l) f
For 1
where 1 <
S
Let ~ be a d-flat in PG(t,p), where q=pn.
Then,
0;
S m ~ v-I, 9~{~m) = 0 far all d-flats ~ if Dq(m(q-l»=s{q-~~
< d.
The functions Bs(t,q) and Rd(t,q) are defined by equations (2.7.10)
50
and (2.7.11), respectively.
From part (ii) of the above theorem, the
number of integers m, 1 ~ m ~ v-I, such that 6E(~m) = 0 for all d-flats
E is at least equal to Rd(t,q).
From the corollary 2.7.1, we have the
following theorem.
Theorem 2.7.5
The rank of the incidence matrix of points and d-flats
in PG(t,q) is at most equal to v - Rd(t,q).
Theorem 2.7.4 may be extended in the following manner.
Since the
coefficients of 6 (X) are either 0 or 1, then, in (~(qt+l),
E
pj
pj.
(6 (X)}
= eE(x:- ), J=O,l, •.. ,n-I.
E
.
pjm
m
Thus, if, for some J=O, 1, ••• ,n-l, e (/3
) = 0, then e (/3 )
E
E
may apply Theorem 2.7.4 to pjm, provided 1 ~ pjm ~ v-I.
Suppose
is the p-adic representation of m(q-l).
Let
t
AO = cO+cnq+···+ctnq
.....
t
A l=cn- l+cn- 1+n q+.•• +c n- 1+t n q •
-nThen
where the terms A. contain only powers of pn=q.
l
a
n-l
= cn-l+c n-l+n +•.• +c n-l+tn •
Let
= O.
We
51
Then
Dq(m(q-l»
Dq (pm(q-l»
= ao+~p+···+an_lpn-l
= an- l+aOP+••• +an-2pn - l
n 1
Dq(P - m(q-l»
= al+~p+···+an_lPn-2 +aOpn-l
•
We see from the above that if the residue of pjm(q-l) mod qt+l_ l
= Dq(u.).
J
is u., then D (pjm(q-l»
J
then
q
e1:(~pjm) = 0
if Dq(pjm(q-l»
Since
6~(~jm(q-l» = e~(aUj),
L.
= s(q-l),
L.
where 1 < s < d.
Hence,
we have
Lemma 2.7.8
If, for some j=O,l, ••• ,m-l, D (pjm(q-l»
q
1 ~ s ~ d, then 61:(~m)
=0
= s(q-l),
where
for all d-flats in PG(t,q), for 1 ~ m ~ v-I.
The condition on m in this lemma is also necessary for e1:(~m) to
be zero for all d-flats 1:, as we shall now show.
shall show that if Dq (pjm(q-l»
More precisely, we
> d(q-l) for each j=O,l, ••• ,n-l, then
there exist integers kO,k l , ••• ,kd such that 0 < kv(q-l) ~ m(q-l),
v = O, ••• ,d, such that kO(q-l) + ••• + kd(q-l) = m(q-l) and
Dp(m(q-l»
= Dp(kO(q-l»
+ ••• + Dp(kd(q-l».
It will follow from
Lemma 2.7.4 that the multinomial coefficient
is not divisible by p and hence e(~m)
Suppose D (pjm(q-l»
q
= b.(q-l),
J
f:
0 for some d-flat 1:.
where b. > d+l, j=O, ••• ,n-l.
J -
recalling equations (2.7.11), (2.7.12) and (2.7.13),
bO(q-l) = Dq(m(q-l»
b1(q-l)
= Dq(pm(q-l»
= aO+alP+ ••• +an_lpn-l
= PbO(q-l) - an_l(q-l)
The~
52
bj(q-l)
from
= pb.J- l(q-l)
- an-J.(q-l)
~ihich
a
o
= pbn-l
- b
O
•
Thus, bO' b , ••• , b _ are restricted to
l
n 1
va~Qes
for which
aO,a1, ••• ,an _l lie between 0 and (t+l)(p-l).
Taking subscripts mod n, for fixed i,
Since
we may write
Ci+Ci+n+···+ci+tn
= pbn_i_l-bn_i'
for which we obtain
c.+c.
. 1
l l+n +••• +c.l+t n + b n-l. + b n-l-
= b n-l. l(P-l).
Let s be the smallest integer such that
(bn-l. l-l)(P-l) < C.-1c.
. l(P-l).
l l+n +•.• ~.l+sn +b.
n-:l- 1 -< b n-l.
Then
(b n-l. 1-2)(p-l) < c.+c.+
l l n +••• +c.l+ ( s- 1) n +b n-l.-bn-l'+1 .
Let
A.
l
I
53
- (c.+c.
.1)] qS ,
1 1+n +... +c.1+ ( S- 1) n +b n-1.-b n-1A~' = [C.+C.
1
. l-(bn-i l-l)(P-l~
1+n+••. +c.+
1 sn +b n-1.-bn-1-
1
Now, A! and A"
contain only powers of pn=q and both are positive
i
1
integers, provided b
i.
S
q
. 1 > 2, which will be the case for at least one
n-1- -
MJreover,
Ai -- A'i +
A~
1
I
and
Dq (A!)
= DP (A!)
= (bn-1. l-l)(P-l)+bn-1.-bn-1. 1 = a.-(p-l),
1
1
1
D
Let
q
(A! I) = D (A! I) = p-l.
1
P 1
o = AO' + Ai'p +••• +An_i'pn-l and Wl=AO + Aip+••• +An_ipn-l .
W
Taking residues mod q-l,
Thus, there exist positive integers, k
WO=kO(q-l) and wl=kl(q-l).
O
and k
l
such that
It is easy to verify the following.
m(q-l) = kO(q-l)
+ kl(q-l)
and
Dp (m(q-l»
We observe that D (k (q-l»
q
1
= Dp (kO(q-l»
+ Dp (kl(q-l».
= a +a p+ •.• +a
pn-l_(q_U = (b -l)(q-l)
0 1
n-l
0
Dq (pn-lkl(q-l»
= (bn- l-l)(q-l)
We may thus repeat the above procedure, with k
1
replacing
ill,
etc.
At
the dth stage, we have obtained d+l positive integers, kO' k1, ••• ,kd
with the required properties.
Hence we have the following theorem.
54
Theorem 2.7.6.
Let 1: be a d-flat in PG( t, q), where q=pn.
(i)
(ii)
e1:(I)
Thl~n
r 0;
m
For I .::: m .::: v-I, e1:(t3 ) = 0 for every d-flat 1: if and only if
for some j=O,I, ••• ,n-l,
D (pjm(q-l»
q
2.8 Incidence matrices of points
~
= s(q-l), where I .::: s .::: d.
d-flats in
EG~.
As in Section 2.6, we shall consider separately the incidence
matrix of points and d-flats of EG(t,q) which pass through the origin,
and the incidence matrix of points and d-flats whieh do not pass
through the origin in EG(t,q).
Here, I.::: d .::: t-l.
The affine case
differs only slightly from the projective case, treated in Section 2.7.
Let 1: be a d-flat in EG(t,q) which passes through the origin.
a be a primitive element of GF(qt).
Let
Then 1: is given as the set of all
those points, aU, of GF(qt) such that
u
e
e
e
(2.8.1)
a = a l a l + a2 a 2 +••• + ad a d ,
where a
el
, ••• , a
ed
t
are independent element s of GF (q ) and where
al, ••• ,a run independently over the elements of GF(q), not all zero.
d
o is also on
1:.
In [24], it is shown that, starting with an initial d-flat 1:,
passing through the origin, the transformation
u
u+l
a ... a
o ...
0
on the points of 1: yields another d-flat passing through the origin.
We may thus generate d-flats from 1: by increasing the exponent of aby
1,2, ••• in the representation of the points of 1:, other than the origin.
The cycle of the flat 1: is shown in [24] to be at most v'=(qt_l)/(q_l).
55
If the points on k, other than the origin, are represented by
u
a l , ... , a
~-l
, where the number of points on a d-flat is k
=
d
q , we
define the incidence polynomial of k as
ul
~-l
9 ()
x =x
+ ••• + x
k
We observe that, over GF(qt), 9 (1) = qd_l
E
91:(1)
f
=-1,
mod p.
Hence,
0.
Let m be an integer such that 1
~
t
m ~ q -2.
Then
+••• +
Since Om = 0, for m > 0, we may consider the summation to be taken over
all possible choices of ~, a , ••• ,ad as elements of GF{q).
2
In an
analogous development of equation (2.7.7), we obtain
(2.8.2)
where the summation is taken over all choices of kl, ••• ,k
° <k)q-l) ~m,
Thus, unless m
= 0,
v=l, ••• ,d,
d
such that
\(q-l) +... +kd(q-l) =m.
mod q-l, 91:(cJfi) = 0.
Moreover, as was argued
in Section 2.7, as necessary and suficient condition that 9 (aF) = 0,
E
for all d-flats 1: passing through the origin in EG{t,q), is that each
of the multinomial coefficients in equation (2.8.2) is divisible by
p, Le., is congruent to zero mod p.
Replacing m by m'{q-l), we have an exactly analogous situation as
in the case of Cd-l)-flats in PG(t-l,q) and the following two theorems
may be similarly proved.
Theorem 2.8.1
Let 1: be a d-flat passing through the origin in EG(t,p);
56
(ii)
For 1
(iii)
For
:s m :s qt_2,
m = m'(p-l),
if m fa 0, mod p-l, 6r.,(r?)
where 1
:s m' :s v'_l,
=
0;
6r.,(aF) = 0 for every d-flat
r., passing through the origin if and only if Dp(m'(p-l»
1 < r < d-l.
Here, v'
Theorem 2.8.2
where q
n
= p.
(i) 6r.,(1)
f
=
=
r(p-l), where
(pt_l)(P_l).
Let r., be a d-flat passing through the origin in
0;
For 1 .::: m .::: qt -2, if m F 0, mod q-l, 6r.,(r?) :: 0
(iii)
For 1 .::: m S qt_2, if m = m'(q-l), where 1 .::: mr
.:::
v'-I, 6r.,(am) =0
for every d-flat r., passing through the origin if D(1 (m'(q-l»
where 1 < r < d-l.
n
where q=p.
(ii)
(iii)
Here, v'
= r(q-l),
= (qt_l)/(q_l).
Let r., be a d-flat in EG(t,q), passing through the origi~
Theorem 2.8.3
f
q),
Then
(ii)
(i) 6r.,(1)
]~G( t,
Then
0;
For 1 S m S qt_2 , unless m e 0, mod q-l, 6r.,(aF) =
o.
t
For lSmSq -2, let m =m'(q-l), where l~~m' ':::v'-l.
Then
8r.,(c?) = 0 for every such d-flat r., if and only if, for some
j=o,l, ••• ,n-~ D (pjm'(q-l»
q
=r(q-l),
where 1 < r < d-l.
-
-
From Section 2.7, the number of integers m, 1 S m S v'-l, such
that Dq(m(q-l»
= r(q-l) where 1 .::: r .::: d-l is equal to Rd _l (t-l,q),
defined by equations (2.7.10) and (2.7.11).
From the above theorems,
it may be shown that the following theorem is true, in an analogous
manner as that used in Section 2.7.
Theorem 2.8.4
Over GF(q), the rank of the incidence matrix N of points
other than the origin and d-flats passing through the origin in EG(t,q)
is at most equal to v' - Rd_l(t-l,q).
In the case q
= p,
this is
57
exact, i.e., the rank of N is equal to v' - Rd_l(t-l,P).
The case of d-flats in EG(t,q) which do not pass through the
origin has been partially treated by Weldon [30].
For completeness,
we shall repeat some of the results found in [30].
Suppose E is a d-flat of EG(t,q) which passes through the point
a C and which belongs to the same parallel bundle as the flat through
e
e
the origin with defining points a 1, .•. , a d.
Then the points on E
are given by those elements aU of GF(qt) such that
where al, ••• ,ad run through the elements of GF(q).
The incidence polynomial of E is the polynomial 9 (x), defined as
E
where the summation is taken over all integers u, 0 < u
that
~
t
q -2, such
aU is given by equation (2.8.3).
Since there are qd points on E, 9 (1)
E
For 1
~
t
m ~ q -2,
=
0, over GF(qt).
58
where the inner summation is taken over all
~
such that jO+jl+••• +jd = m and 0
jy
~
choice~s
of jo' jl"'" jd
m, V=O,l, .•. ,d
Interchanging the order of summation and noting that, for v=l, ••. ,d,
=
-I, if j y
{
= k y(q-l),
k y > 0,
0, otherwise,
we have
(2.8.4)
x
where the summation is taken over all choices of jo' ky(q-l), v=l, ••• ,d,
such that 0 ~ jo ~ m, 0 < ky(q-l) < m, v-l, ••• ,d, and
jo + kl(q-l) +••• + kd(q-l)
= m.
If each of the multinomial coefficients in equation (2.8.4) is
divisible by p, then 9~(aF) = 0, for every d-flat ~ not passing
through the origin in EG(t,q).
If one of these multinomial coefficients is not divisible by p,
it is not divisible by q = pn.
This will hold if and only if
Dq(m) = Dq(jo) + Dq(kl(q-l)) + .•• + Dq(kd(q-l)).
Since Dq (jO) -> 0 for jo -> 0 and for v=l, ••• ,d, Dq (k y (q-l)) _> q-l,
equation (2.8.5) holds only if D (m) > d(q-l).
q
-
Thus, if D (m) < d(q-l),
q
each of the multinomial coefficients must be divisible by q and hence
by p.
We have the following theorem.
Theorem 2.8.5
If
the origin, then
(i)
e~ (1) = 0;
~
is a d-flat in EG(t,q) which does not pass through
59
For 1 ~ m ~ qt_2, if Dq{m) < d{q-l), then 9 {aF) = 0 for all such
k
(ii)
d-flats k.
Recall that the function Bs{t,q), defined by equation (2.7.10), is
the number of ways of choosing t+l integers c ,c , ..• ,c such that
t
O 1
o ~ ci
~ q-l, i=O,l, ... ,t, and c +c +... +c = s{q-l).
t
O 1
t
For 0 ~ m ~ q-4
let the q-adic representation of m be
where 0 -<A.J .<- q-l, i=O,l, ••• ,t-l.
Then
Since the q-adic representation of m is unique, the number of
integers m, 0 < m < qt_ 2, such that D (m) < d(q-l) is equal to the
-
-
q
number of ways of choosing t integers A ,A , .•• ,A _ such that
t 1
O l
o~
Ai ~ q-l, i=O, ••• ,t-l, and AO+Al+ ••. +At _l < d(q-l).
Let this
number be Kd(t,q).
Let Ds(t,q) be the number of ways of choosing t integers
AO"."~_l such that 0 ~Ai ~ q-l and (s-l) (q-l) ~AO+ ••• +At_l<s(q-l).
Then
d
(2.8.6)
=
Suppose that AO+ ••• +A t _l
I
D
s
= s(q-l)
(t,q).
- Et , where 0 < Et ~ q-l.
A +A +·.·+At _ + Et
1
o l
The number of integral solutions
Then
= s(q-l).
Ao, ... ,At_l,E t
of this equation such
that 0 ~ Ai ~ q-l, i=O, 1, ••• , t-l and 0 ~ Et ~ q-l is equal to Bs (t, q).
But this number also includes those solutions A , ••• ,At _1,E t such that
O
E = 0, i.e., such that AO+A1+ ••• +A _
t 1
t
such solutions.
= s(q-l).
There are Bs (t-l,q)
Hence the required number of solutions is
60
Ds (t,q) = Bs (t,q) - Bs (t-l,q) .
Substituting this into equation (2.8.6), we obtain
d
I
d
Bs(t,q)
-
s=l
I
B (t-l, q) .
s
s=l
But
Bs (t,q) •
Therefore,
Combining the above results with Theorem
Theorem 2.8.6
The number of integers m, 0
~
2.8.~5,
we have
t
m ~ q -2, such that
91':(~) = 0 for every d-flat 1': not passing through the origin in
EG(t,q) is at least equal to Kd(t,q).
As a corollary, it may be shown that the rank of the corresponding
incidence matrix of points other than the origin and d-flats not
t
passing through the origin in EG(t,q) is at most equal to q -l-Kd(t,q).
Analogous to Theorem 2.8.3 is the following theorem.
Theorem 2.8.7
the origin.
(i)
(ii)
91:(1)
Let 1': be a d-flat in EG(t,q) which does not pass through
Then
=0
For 1 ~ m ~ qt_2 , if, for some j=O,l, •.• ,n-l, Dq(pjm) < d(q-l),
then 91:(~)
=0
for all such d-flats 1':.
CRAPI'ER III
GEOMETRIC CODES
3.1 Introduction
A number of classes of cyclic codes to which a majority decoding
algorithm is applicable may be given a geometric interpretation.
shall call such codes geometric codes.
We
Included in this grouping are
the Reed-Muller codes and some of their generalizations.
We shall
define several new classes of geometric codes and investigate some of
their properties, particularly as related to the BCH codes.
A majority
decoding algorithm for the geometric codes will also be given.
3.2 Cyclic codes
A familiarity with the theory of linear and cyclic codes will
be assumed in this chapter.
We state here a few of the specialized
results on cyclic codes which are needed later.
For a more complete
description of cyclic codes, see Peterson [21], for example.
A q-ary linear code C is a vector subspace of the vector space
VN of all N-vectors with elements from GF(q); q being a prime power,
say q
n
= p.
The dimension, k, cf the subspace C is called the number
of information symbols of the code C.
C is referred to as an (N,k) code.
space of C by CD.
dual code of C.
If r
= N-k,
The length of the code is N.
Denote the orthogonal or null
CD is an (N,r) linear code called the
The number r is the redundancy of the code C.
62
A matrix G whose row vectors span or generate C is called a
generator matrix of C.
A matrix H whose row vectors span the dual
code is called a parity check matrix of C.
is a code vector of C if and only if c'
€
A vector £'=(cO,cl, ••• ,cN_~
C, which is
tr~e
if and only
if
(3.2.1)
H c
= O.
Each of the equations in (3.2.1) is a parity ~,equation. Indeed,
if £'=(hO,hl' ••• '~_l) is a vector of CD' then ~'yields the parity
check equation
A linear code C (a vector supspace of V ) is cyclic if, for every
N
code vector £'=(cO'cl, ••• ,c N_l ) of C, the vector (CN_l,cO, ••• ,cN_2)'
obtained from £' by shifting each coordinate one unit to the right,
is also a code vector of C.
A convenient represen"tation of cyclic
codes may be made through the theory of ideals in the residue class
ring GF(q) [x]/(XN_l).
In the residue class ring GF(q) [x]/(xN_l), we correspond the
(residue class of the) polynomial c(x) =cO+clx+ ••• +~_lxN-l with the
vector c' = (cO'cl, ••• ,cN_l ).
c(x) is called the polynomial of the
vector £', and £', the vector of c(x).
Under this correspondence, it
may be shown that a linear code is cyclic if and only if it is an
ideal in the ring GF(q) [x]/(xN-l).
Each such ideal C contains a
unique monic generator polynomial, g(x), of smallest degree less than
N such that (the residue class of) c(x)
c(x).
E
C if and only if g(x) divides
Moreover, g(x) is a divisor of xN-l in GF(q), say xN-l=g(X)h(X).
If the degree of hex) is k, then g(x) is of degree r = N-k and the
63
code C is an (N,k) cyclic code.
parity check polynomial of C.
The polynomial h(x) is called the
The dual code of C is also cyclic and
its generator polynomial, denoted gn(x), is given by
and is of degree k.
A cyclic code may be specified by the roots of its generator
m
polynomial in an extension field of GF(q).
u > 1.
Suppose q -1 = Nu, where
(We assume N and q are relatively prime, so that xN_l has no
repeated roots.)
Every root of xN_l is an element of GF(~).
0: be an element of GF ( q ) of order
.
N,~.e.,
0:
Let
N .. l
=1 and 1,0:, cl- , ••• ,or"
N
Then the roots of xN-1 are 1,0:, cl- , ••• ,or"N-l • Any
u
u
u
sub se t 0 f these roo t s, say 0: 1 ,0: 2 , ••. ,0: s , spec~. f'~es a cyc l'~c co d e
are all distinct.
C.
The (residue class of the) polynomial c(x) is a code po1¥nomial
u
1
of C if and only if 0: ,
u
... , 0: S are
roots of c(x).
I f 'u. (x) is the
~
u.
minimum function of a
~
, i. e., the monic polynomial of srrll'tllest degree
u.
less than m in GF(q) [x] which has 0: l as a root, then c(x) is a code
polynomial if and only i f .u. (x) divides c(x), i=l, ••• ,s.
The
~
generator polynomial is g(x) = L.C.M. {+ (x), ••• ,, (x») and is of
ul
Us
degree at most equal to sm; in general, some of the • (x) may be
u.~
identical, in which case the degree of g(x) is less than sm.
The
degree of g(x) is equal to the number of distinct elements in the
ul
qUI
u2 qu...
u
qu
.
the numb er
sequence 0: ,0:
, ... ,0:,0: , ~ , ••• ,0: s ,0: s , ..• ,l.e.,
0
f
If N = qm_l , i.e., if u=l, then 0: is a primitive element of GF(qm)
64
and the code C is called a primitive
If u > 1, then a
cyclic~.
is not a primitive element and the code is a non-primitive cyclic
cod~
Every element of GF(qm) may be represented as a pmver of a
primitive element,
~,
or as a polynomial in
)
with coefficients in GF ( q.
Suppose
u
~
of degree at most m-l
= bO + bl~
~
"m-l
+••• + bm_IP
.
Corresponding each such element with the column vector of coefficients,
it may be verified that the matrix H, given by
H
u
N-l
u
N-I
I
(a 1)
I
(a 2)
,
=
u
1
u
a s
(a s)
N-l
when reg::1rded as a sm XN matrix over GF(q), is a parity check matrix
u
u
of the code C, specified by a 1, ••• , a s as roots of every code polynomial.
Associated with the cyclic (N,k) code C is an extended cyclic
code, Ce , which is not a cyclic code.
If the generator polynomial of
C does not have I as a root, then C is defined I'l.t the linear code
e
with parity check matrix
1
H
e
o
o
=
III
H
o
In this case, C
e
is a (N+l,k) code.
If 1 is a root of ~(x),
65
u
suppose a
l
1, then C is the linear code with parity check matrix
e
1
,
o
=
He
o
H
I
o
.J
in I'lhich case, C is a (N+l,k+l) code.
In either case, the c;enerator
e
polynomial of the code C will also be referred to as the generator
polynomial of the extended code C •
e
The best cyclic codes yet discovered are the well lmovrn BoseChaudhuri-Hocquenghem (BCH) codes [5], [6], [13], which are most
easily defined in terms of the roots of the generator polynomial.
For any integer m and any prime power
element of GF{qm) of order N.
m
q -1 = Nu.
q
let a be a non-zero
m
N must be a divisor of q -1, say
Let c and d be integers, ?
~
d
~
N-2.
The q-ary cyclic
code whose generator polynomial, g{x), has roots
c
a,
a c+l , ... , ac+d-2
(3.4.1)
is a
~
code.
Usually, c is tru:en as 1.
In the case u=l, the code
is a primitive BCH code; otherwise, it is a non-primitive BCH code.
.
The matrix
i
H
=
1
{ac)N-l
1
{ac+l)N-l
,
66
when considered as a (d-l) X N matrix over GF(qmL may be shown to
hAye the property that no d-l columns of H are dependent over GF(qm).
Since H may also be regl'trded as a m(d-l) X N pl'trity check matrix over
GF(q) and no d-l columns of H are dependent over GF(q), then the code
has mininmm distAnce at least d.
For this reason, the number d is
often referred to as the designed distance of the BCH code specified
by equation (3.4.1).
If d=2t+l, i t is possible to correct up to t
errors.
Since the rank of H, over GF(q), is at most m(d-l), the redundancy of the BCH code is at most m(d-l) and the number of information
symbols is at least N-m(d-l).
The I'tctual redundancy is equal to the
degree of g(x), the generator polynomial, and may be less than m(d-l),
depending on the degrees of the mininrum functions of the roots in
(3.4.1).
An investigation of the number of infor~,tion symbols in
BCH codes may be found in [2] and in [17].
In some C1".ses the true minimum distance of a BCH code may exceed
the designed distance.
For example, the Golay (23,12) code is Fl.
non-primitive BCH code for which the designed distance is 5, but for
which the actual distance is J:nOl'ffi to be 7 [19], [?l].
Several
authors have studied the distance properties of BCH codes for particular values of the parl'tmeters [14], [15], [19], [29]; most BCH codes
are found to hFlve their true mininmm distances equal to the designed
distance of the code.
The most important BCH codes are the primitive binary BCH codes,
obtained by setting n=2, u=l, c=l and d=2t+1.
If ex is a primitive
m
element of GF(2 ), the generator polynomial g(x) of the corresponding
BCH code has roots a,
2
or,
... , or2t-l . If t.(x)
is the
1
function of ai, then t J..(x) = '.(x).
1
2
and has degree at most equRl to mt.
minimum
g(x) is given by
m
The length of the code is N=2 -1
m
and the number of information symbols is at leRst 2 -l-mt.
The code
will correct t or fewer errors vath certainty.
3.3 Majority decoding
A relatively simple decoding procedure, called majority decoding,
is applicable to some lineRr codes.
A comprehensive study of majority
decodinf; and more generally, threshold decoding, particularly as
applied to convolutional codes, is given by Massey [18].
We shall
restrict our discussion to majority decoding of linear and cyclic
codes.
The concept of majority decoding of linear codes is based on a
certain 'orthogonality structure' the parity check equations may have.
We first define this orthogonality structure and show
hOVI
it relates
to majority decoding.
Let C be a (N,k) linear code with symbols from GF(q).
C is
completely specified by a set of parity check equations, which need
not be linearly independent.
check
eqL~tions
for C is
Suppose a set of r' > r=N-l: parity
68
A vector c'
(cO'cl, ... ,c _ ) is a code vector of C if and only
N 1
=
if the coordinates of £' satisfy the system of equations (3.3.1).
f
jth equation is said to check the symbol d. if h ..
1
Jl
O.
Tl'e
A subset of
J equations, say the first J equations in (3.3.1), is said to be
orthoGonal
~
the symbol c i [18] if
(1)
h .. = 1, j=l,2, •.• ,J, and
(2)
h
f
i.
any fixed it
Jl
ji
,= 0 for all but at most one j=l,2, .•• ,J and for
That is, the first J equations are orthogonal on c. if each of the
1
first J equations checks c. ,-lith coefficient 1 and no other symbol is
1
checl\:ed by more than one of these J equations.
For example, in the system of equations
=0
=0
=0
C:z
')
+ c
+ c4 + c 5
c
+ c
l
+ c
= 0
5 + c6
+ c6
2
o
+ c6
4
:= 0
= 0,
the first, fifth and seventh equations are orthogonal on cO; the
first, second and sixth are orthOGonal on c ' etc.
l
Let us represent the system of equations (3.3.1) in matrix form
as
H c
=
0,
where H is a r t X N parity check matrix of rank rover GF(q).
69
Suppose
i' = (to,tl, •• ·,tN_l )
is a transmitted code vector of C
and the corresponding received vector is £' = (rO,rl, ••• ,t _ ).
Nl
.
error vec t or lS
syndrome of
Eo
have
~
' -- ( eO,el, •.• ,e _ ) , were
h
£ ' - _t' +
Nl
£' is
= H ~
~
= H £.
'
~.
The
The
Since t' satisfies equations (3.3.2), we
which we may vrri te as
It is desired to solve the system (3.3.3) for the (unknovm)
error vector, given the matrix H and the syndrome vector
s'
= (sl, ••• ,sr').
This is called the decoding of the code C.
If it is possible to find a set of J parity check equations from
(3.3.3) which are orthogonal on the symbol eO' for example, then a
majority decoding algorithm for determining eO is given by the
follovdng theorem, due to Massey [13].
Theorem 3.3.1.
Suppose
(3.3.4)
is a system of J equations orthogonal on the symbol eO.
that
[J/2J 1 or fewer of the symbols checked by these
Provided
J equations
are non-zero, then eO is given correctly by the following rule:
(1)
eO is that value of GF(q) which is assumed by the greatest
fraction of the (Sj)' provided such a value exists.
1
The notation [x] denotes the greatest integer less than or equal
to x.
70
(?)
In the case where no single value is assumed by a strict
plurali~
of the (Sj)' eO is zero.
This theorem is proved in [18].
An examination of the proof as
given in [18] suggests that the condition of the orthogonality' of
the equations
(3.3.4) on eO is not necessary, as we shall prove in
the follovnng theorem which is a generalization of the above; this
[26].
next theorem appears also in Rudolph
Theorem
3.3.2 Suppose that each of the equations (3.3.4) checks the
f
symbol eO and assume that, for any i
of these J equations.
Let A.
J
0, e i is checked by at most ~
= s./h. O'
J
J
Then, provided at most [J/2~]
of the symbols checked by these J equations are non-zero, eO is given
correctly by the following rule.
(1)
eO is that value of GF(q) 'Vlhich is assumed by the greatest
fraction of the (A ), if such a most frequent value exists.
j
(2)
In the case where no single value is assumed by a strict :plurality
of the (A ), eO is zero.
j
Proof.
Suppose at most t
equations
=
[J/25] of the symbols checked by the
(3.3.4) are non-zero. First, assume that all symbols other
than eO are zero.
Then
and the decision rule of the theorem is correct.
symbols is non-zero, then at most
different from eO'
of the A of equations
j
(3.:5.5)
In general, if s other symbols are non-zero, at
most s5 of the A of equations
j
~
If one of the other
(3.3.5) are different from Aj' since
each such symbol can affect at most
~
of the A ..
J
are
71
If eO
most t5
=
0, then if t or fe,,,er of the symbols are non-zero, at
= [J/2] of the A.J are non-zero and the decision rule of the
theorem gives the correct value of eO.
If eO
f
0, then at most
(t-l)5 < [J/2] of the Aj are not equal to eO and the decision rule
of the theorem is correct.
We observe that Theorem 3.3.1 is a particular case of Theorem
3.3.2 Hith 5
= 1.
The decoding rule of Theorem 3.3.2 will be re-
ferred to as majority decoding.
It should be remarked that a set of
J equations satisfying the conditions of Theorem 3.3.2 must be available for each e., i=O,l, ••• ,N-l for complete decoding of an error
~
vector.
More accurately, only the information symbols in an error
vector need be decoded if the encoding procedure is available to
reconstruct the remaining symbols of the corresponding codew·ord.
For the case of a cyclic code, there is a 'symmetry' which ensures
that if one symbol may be decoded by majority decoding, then all the
other N-l symbols in a codeword may also be so decoded.
The majority decoding algorithms of Theorems 3.3.2 and 3.3.1
are both one-step majority decoding procedures, i.e., only one determination of the majority of the A. is necessary to decode a
J
particular symbol.
~symbols,
and e
l
The algorithm may be used to decode certain sums
such as eO+e , provided J equations which check both eO
l
may be found.
Suppose a number of sums involving eO may be
decoded in this manner.
These decoded sums may be treated as new
parity checks and the majority decoding procedure could be applied
to these new equations to decode eO'
Such a decoding procedure is
referred to as 2-step majority decoding, or in general, L-step
majority decoding.
In [18], examples are given of particUlar codes
72
which may be decoded in this manner.
The well known binary Reed-
Muller codes are such a class of codes.
Many of the codes to which
this majority decoding procedure is applicable
~y
be given a geometrk
interpretation and are generalizations of the Reed-Muller codes.
This
will be discussed in the next sections.
3.4 Geometric codes
Underlying the (cyclic) Reed-Muller codes [20],[25], [15J and
some of their generA.lizations is a geometric interpretation of the
dual code.
In this section vle shall examine some of these cod.es in
this light and in Section 3.5, indicate how such a geometric interpretation yields a majority decoding algorithm.
Let
a be a primitive element of GF(qt) and let C be a primitive
cyclic cod.e of length qt -1 \-lith symbols from GF( q).
generator polynomial of C by g(x).
C is a (qt_l,qt_ l _r ) code.
=
(x
then gn(X) is of degree k
gn(x)
If the degree of g(x) is r, then
Let the dual code of C be Cn' Hith
generator polynomial gn(x).
hex)
Denote the
q
If hex) is given by
t
-l_l)/g(x),
= ot_ l _r
and is given b'y
= xkh(x- l ).
We ITlAy completely specify C by specifying the roots of gn(x) in
GF(qt).
Since the degree of gn(x) is the number of information symbols
of C, it is desirable that gn(x) have a rell'l,tively Ip.rge number of
roots.
Let E be a ~-flat in EG(t,q) VTith incidence polynomial 6 (x),
E
defined in Section 2.8.
not be accounted for.
Unless otherwise specified, the origin will
The incidence vector of E is the (qt-l)-vector
73
of coefficients of 9~{X).
This incidence vector is a vector of the
dual code CD if and only if 9r,(X) is divisible by gD(x).
that if
a.o
=
We observe
I is a root of gD (x), then the dual code cannot contain
the incidence vector of any flat in EG{t,q) which passes through
the origin, for 1 is not a root of the corresponding incidence polynomial, by Theorem 2.8.2.
We shall refer loosely to a primitive cyclic code of length
qt_l with symbols from GF{q) whose dual code contains the incidence
vectors of either all ~-flats in EG(t,q) or all ~-flats of EG(t,q)
which do not pass through the origin as a primitive geometric code
of order.l:!'
In the case the dual contains the incidence vectors of
all
we shall refer to the code as a primitive geometric code
~-flats,
Q; otherwise, if the dual contains only the incidence vectors
of
~
of
~-flats
which do not pass through the origin, we shall refer to
the code as a primitive geometric code of
For the non-primitive case, let
order N=(qt+l=l)/(q_l). We may take ~
element of GF(qt+I).
~
~
1,
as necessary.
be an element of GF{qt+l) of
= aq- \
where a is a primitive
Let C be a non-primitive cyclic code of length
N with symbols from GF(q) and let the generator polynomial of the
dua.l code CD be gD (x).
Then the dual code containS>' the incidence
vector of the p-flat ~ of PG(t,q) if and only if the incidence polynomial 9r,(X) of T., defined in Section 2.7, is divisible by gD(X)'
We shall refer to a non-pri.miti ve cyclic code of length N whose dual
contains the incidence vectors of every ~-flat in PG(t,q) as a nonprimitive geometric
~
£f
order
~.
74
Theorem
(1)
3.4.1
t
A sufficient condition for a cyclic code of length q -1 to be a
type 0 primitive geometric code of order ~ is that every root
of
of
the generator polynomial gn(X) of the dual code satisfy
t
(3.4.1)
1 :::: h :::: q -2
for some j
= O,l, .•• ,n-l.
(2)
0 < n (pjh) < ~(q-l)
and
q
t
A sufficient condition for a cyclic code of length q -1 to be a
type 1 primitive geometric code of order ~ is that every root
of
of
the generator polynomial of the dual code satisfy
0:::: h:::: qt_2 and n (pjh) < ~(q-l) for some j=O,l, •.• ,n-l.
(3.4.2)
q
Proof.
If a root
Theorem
2.8.3,
~-flat
or of gn(x)
satisfies equation
or is also a root
(3.4.1), then, from
of the incidence polynomial of every
in EG(t,q) which passes through the origin.
From Theorem
2.8.4, ~ is also a root of the incidence polynomial of every ~-flat
which does not pass through the origin.
the indicence vectors of all
~-flats
in EG(t,q), i.e., the code is a
type 0 primitive geometric code of order
If
of
satisfies equation
Hence the dual code contains
~.
(3.4.2), then
or is also a root
of the
incidence polynomial of every ~-flat in EG(t,q) which does not pass
through the origin, from Theorem
geometric code of order
Theorem
~
2.8.7, and the code is a primitive
and type 1.
3.4.2 A necessary and sufficient condition for a non-
primitive cyclic code of length N
=
(qt+l_l)/(q_l) with symbols from
GF(q) to be a non-primitive geometric code of order
~
is that every
root f3h of the generator polynomial gn(x) of the dual code satisfy
75
1 < h < N-l and 0 < D (pjh(q-l)) < ~(q-l)
q
for some j
Proof.
= O,l, .•• ,n-l.
From Theorem 2.7, 6,
11
~
is a root of the incidence polynomial
of every ~-flat in PG(t,q) if and only if h satisfies equation (3.4.3).
The most well known geometric codes are the (binary) ReedMuller codes, of which the duals of the distance 3 and distance 4
HamminG codes are particular cases.
The Generalized Reed-Muller
codes of Kasami, Lin and Peterson [15] will be shown to be primitive
geometric codes.
Weldon's Difference-Set Cyclic codes [28] are
examples of non-primitive geometric codes.
We shall briefly describe
these and other geometric codes here and then define new classes of
both primitive and non-primitive geometric codes.
A summary of these
codes is given in Table 3.1.
With the exception of the Hamming codes, each of the above codes
may be described conveniently in terms of the roots of the generator
polynomial of the dual code.
Definition 3.4.1
distance
l
t
Let a be a primitive element of GF(2 ).
The binary
Hamming code is defined as the cyclic code of length
t
N = 2 _l with symbols from GF(2) such that c(x) is a code polynomial
if and only if a is a root of c(x).
Definition 3.4.2
Let a be a primitive element of GF(2 t ).
The binary
distance ~ Hamming ~ is defined as the cyclic code of length
N
= 2 t _l
with symbols from GF(2) such that c(x) is a code polynomial
if and only if 1 and a are roots of c(x).
The properties of the Hamming codes are well known and are
76
discussed in Peterson [21].
We mention here that the generator poly-
nomial of the distance 3 Hamming code has roots precisely those
elements
cP)
1'::: h .::: 2t -2 such that D (h)
2
= L
~The generator poly-
nomial of the distance 4 HamminG code has as its roots precisely those
11
t
elements ci) 0'::: h .::: ? -2) such that D2 (h) .::: 1.
Definition 3.4.3
[15] Let abe a primitive element of GF(2 t ).
v-th order Reed-Muller code is the cyclic code of length N
The
= 2 t -1
with symbols from GF(2) such that the generator polynomial gD(x) of
the dual code has as roots precisely those elements
or)
t
0 .::: h .::: 2 _2)
This definition of the Reed-Muller codes is shown in [15] to be
equivalent to that of the original Reed-Muller codes [20], [25]) with
the first coordinate deleted.
We shall refer to the original Reed-
Muller codes as the extended Reed-Muller codes) for it is shown in
[15] that these codes are extended cyclic codes.
The modified Reed-Muller ~ £! order v is the
t
cyclic code of length N=2 _l whose dual code is obtained from the
Definition 3.4.4
dual code of the vth order Reed-Muller code by deleting the element
1 as a root of the generator polynomial.
The generator polynomial of the dual code of the modified ReedMuller code has as roots those elements
J1
t
such that 1 < h < 2 _2
Let us denote the vth order Reed-Muller code by C and the dual
v
t
,
.
t
h
code by CD • For O .::: h .::: 2 - ..., vTrl e
= Co +clr.+••• +c t _l 2t -l' where
, v
r)
77
the c
i
are 0 or 1.
We see that the degree of gD(X) is
k=l+t+(~)+ ...
is a (2 t _l,k) code.
v
g(x) = xr~(x-l) where
So C
2
and is of degree r = 2 t _l_k.
t
D (2 -I-h)
2
=t
.:s h .:s 2 t -2
0
- D (h).
2
(~)
The generator polynomial of C is
v
hn(x) = (x
a -h such that
+
t
1
- -l)/gD(x)
Now ~(x-l) has as roots those elements
and D2 ( h )
~
vorl.
But a-h =
rl-
t
-l-h and
hI
Hence, g(x) has as roots those elements a
such that 1 < hI < 2t -2 and 0 < D (h I)
2
.:s
t- V-l.
From Definition
3.4.4, ue see that the Reed-Muller code of order vis the dual of
the modified Reed-Muller code of order t-v-l.
It follows from Theorem 3.4.1 that the vth order Reed-Muller
code is a type 1 primitive geometric code of order vorl Rnd that the
modified vth order Reed-Muller code is a type 0 primitive geometric
code of order v+l.
The preceeding results on the Reed-Muller codes are well l,novffi,
but not generally as a result of the notion of a primitive geometric
code.
We may mention here that the distance 3 Hamming code is the dual
code of the mofified first order Reed-Muller code and the distance 4
Hamming code is the dual of the first order Reed-Muller code.
The following generalization of the Reed-Muller codes to the
non-binary case is due to Kasami, Lin and Peterson [15].
Difinition 3.4.5
t
Let a be a primi ti ve element of GF (q ).
The vth
order Generalized Reed-Muller (GRM) ~ is the cyclic code of
78
length N
= qt_ l
with symbols from GF(q) such that the generator
polynomial gD(x) of the dual code has as roots those elements
oF,
0 < h < qt_2, such that D (h) < v.
-
-
q
It is shown in [15] that the generator polynomial g(x) of the
vth order GRM code has as roots those elements
aP,
1 < h < qt_2,
such that 0 < D (h) < t(q-l) - v-I.
q
-
We may define a modified GRM code by deleting the root 1 as a
root of the generator polynomial of the dual code.
MOre precisely,
we have
Let ex be a primiti ve element of GF (qt).
Definition 3.4.6
modified vth order GRM
~
~rhe
is defined as the cyclic code of length
at_l with symbols from GF(q) such that the generator polynomial of
the dual code has as roots those elements
ap,
1 ~ h ~ qt_2 , such
that 0 < D (h) < v.
q
-
Thus the vth order GRM code is the dual of the
GRM code, where ~
= t(q-l)
Lemma 3.4.1 For v
=
~th
order modified
- v-I.
J,l(q-l) + s, vThere 0 ~ s < q-l, the vth order
GRM code is a type 1 primiti ve geometric code of' order
~
Rnd the
vth order mdified GRM code is a type 0 primitive geometric code of
order J,l H.
Proof.
If Dq (h) ::: v, then Dq (h) < J,l(q-l) and the lemma is irmnediate
from Theorem 3.4.1.
We now define a now class of primitive geometric codes, which
we call the Affine Geometry codes.
We use the term Affine Geometry
79
because the term Euclidean Geometry has been applied to a class of
codes by Weldon [30]; we shall discuss these codes later.
Definition 3.4.7
The
~th
Let a be
R,
primitive element of GF(qt), where a=pn
order Affine Geometry code is the primitive cyclic code of
length qt_ l with symbols from GF(q) such that the generator polynomial gD(x) of the dual code has
o~
t
aP
as a root if and only if
.
h ~ q -2 and Dq (pJh ) < Ji(q-l) for some j=O, 1, ••• ,n-l.
This specification of the roots of the polynomial gD(x) is
valid, for ifah satisfies D (pjh) < ~(q-l),
q
then Dq (pjh)
= Dq (pjh) < ~(q-l).
Thus the conjugates a
qh
, •.• , of
are also included in the specification of the roots of gD(x) if
or
or
is a root of gD(x).
As an immediate consequence of Theorem 3.4.1, we have
Lerrnna 3.4.2 The Jith order Affine Geometry code is a type 1 primiti ve
geometric code of order
Definition 3.4.8
~.
Let a be
::'I
primitive element of GF(qt).
The
modified lJ,th order Affine Geometry code is the primitive cyclic code
of length qt_l with symbols from GF(q) such that the generator polynomi::tl of the dual code has
aP
as a root if and only if
1 ~ h ~ qt_2 and 0 < Dq (pjh) ~(q-l) for some j
= O,l, ••• ,n-l.
80
That is, the modified
from the
~th
~th
order Affine Geometry code is obtained
order Affine Geometry code by deleting the element 1
as a root of the generator polynomial of the dual code.
Again,
from Theorem 3.4.1, we have
Lemma 3.4.3
o primitive
The modified
= (~(p-l)
order Affine Geometry code is a type
geometric code of order
For the case q
v
~th
= p,
the
~.
th order Affine Geometry code is the
-l)th order GRM code.
In general, the (~(q-l) -l)th
order GRM codes are subcodes of the
~th
order Affine Geometry codes,
as will follow from this next lemma.
Lemma 3.4.4
Let C and C be cyclic codes of length N with symbols
l
2
from GF(q) and let gDl (x) and ~2(x) be the generator polynomials of
the dual codes, respectively.
Then C is a subcode of C if and only
l
2
if gDl(x) divides gD2(x).
Proof.
Let the dual codes be CDI and CD2 , respectively.
C So C if and only if C So C •
Dl
D2
2
l
Then
It is thus sufficient to shaw that
CD2 S CDl if and only if gDl (x) divides ~2 (x).
Suppose gDl(x) divides gD2(x).
Then every code polynomial of
CD2 is divisible by ~2(x) and gDl(x), i.e,
CD2
S
CDl ·
Conversely,
suppose that every code polynomial of C is also a code polynomial
D2
of C •
DI
Since gD2(x) is a code polynomial of CD2 , then gDl(x) is
also a code polynomial of CDI and hence is divisible by gDl'
This
81
completes the proof of the lemma.
Since every root of the generator polynomial of the dual code
of the (~(q-l)-l)th order GRM code is also a root of the generator
polynomial of the dual code of the
~th
order Affine Geometry code,
By
order GRM code is a subcode of the
order Affine Geometry code.
~th
Lemma 3.4.4, the
(~(q-l)-l)th
the one polynomial divides the other.
In
general, the GRM code will be a proper subcode of the corresponding
Affine Geometry code, for there will, in general, be integers h such
that D (h) > ~(q-l) while D (pjh) < ~(q-l) for some positive j.
q
-
q
For example, take the case t = 4, q = 4 = 2
v = (2(4-1)-1) = 5.
and thus D4 (43)
D (86)
4
= 5 < 6.
For h
= 7 > 6.
= 43,
2
and
~
= 2.
Let
2
we may write 43 = 3 + 2x4 + 2x4
However, 2 X 43
= 86 = 2
2
+ 4 + 4 + 43 and
43
Thus, a
is a root of the generator polynomial of
the dual of the 2nd order Affine Geometry code, but is not a root
of the generator polynomial of the dual of the 5th order GRM code
For the case q
= 2n ,
the modified Affine Geometry codes are
subcodes of Weldon's Euclidean Geometry codes [30], which we shall
describe slightly differently than as given in [30].
Difinition 3.4.9
Let a be a primitive element of GF(2
order Euclidean Geometry
~
nt
).
The vth
is defined as the cyclic code of length
82
2nt _1 with symbols from GF(2 n ) such that the roots of the generator
polynomial of the dual code are precisely those elements
that
aP
ah
such
is also a root of the incidence polynomial of every v-flat
In [30], the Euclidean Geometry codes are described in terms of
the parity check polynomial, rather than the generator polynomial of
the dual code; this merely reverses every vector in the dual code.
Because of the difficulties of determining exactly those
elements
~-flats
aP which
are roots of the incidence polynomials of all
in EG(t,q), we have not defined the Affine Geometry codes as
a natural extension of the Euclidean Geometry codes to the case
n
q=p •
Before considering further properties of the
Geometry codes, let us consider the dual codes.
~th
order Affine
We shall show that
the duals of Affine Geometry codes are subcodes of certain BCH codes.
Let
h
O
\..
= q ~ -1 = (q-l ) + (q-1M
+ ... + (q-l ) q ~-l ,
which we may write as
hO = (p-l) + (p-IA
+
+ [(p-l) + (p-l)q +
+ (P_I)q~-l
+ (p-l)q
JJ-l
]p
+
+ [(p-l) + (p-l)q + •.. +
(P_l)q~-l]pn-l.
From the latter representation, we see that for each j=O,l, ••. ,n-l,
Dq (pjh ) = JJ(q-l) and for all h < h ' Dq (pjh) < Dq (pjh) = JJ(q-l).
O
o
ThUS, the generator polynomial of the dual code of the
t
~th
order
Affine Geometry code of length q -1 has at least the elements
83
2
h _
h
Ol
l,ex,a-, •• • ,ex
, but not ex O as roots.
of a
~th
Therefore, the dual code
order Affine Geometry code is a subcode of the BCH code of
length qt_ l with designed distance hO+l = ~ and specified by the
2
qU_ 2
roots l,ex, a-, ... ,ex
.
Then the minimum distance of the dual of
the ~th order Affine Geometry code
js
at least equal to qJ.l.
Since
there are (qt-~_l) • (t-l,J.l-l,q) ~-flats in EG(t,q) not passing
through the origin and the incidence vectors of each such flat have
weight qJ.l, there are at least (qt-J.l_ l ) • (t-l,J.l-l,q) vectors of
minimum weight qJ.l in the dual of the J.lth order Affine Geometry code
and hence also in the above BCH code.
For the modified J.lth order Affine Geometry code, the generator
2
q~-2
polynomial of the dual code has at least the elements ex,a-, ... ,ex
,
qJ.l 1
but not ex - , as roots. Thus the dual code is a subcode of the BCH
code with designed distance qJ.l_l and specified by the roots
2
qJ.l_
ex,a-, ••• , ex 2 • Since there are t(t-l,J.l-l,q) J.l-flats in EG(t,q)
passing through the origin and each of the corresponding incidence
vectors has weight qJ.l_ l , the dual of the modified ~th order Affine
Geometry code and the above BCH code contain at least t(t-l,J.l-l,q)
vectors of minimum weight qJ.l_1 and at least (qt-J.l_l)t (t-l,J.l-l,q)
vectors of weight qJ.l.
Lemma
3.4.5.
We state these results as a lemma.
The BCH code of length qt_l with designed distance
qJ.l_l and specified by the roots ex,
exactly qJ.l_l •
cl-, ... ,
~
exq -2 has mininrum distance
Moreover, there are at least t(t-l J.l-l,q) vectors of
weight q~-l and at le'3.st (qt-J.l_ l ) t (t-l,J.l-l,q) vectors of weight q~
in the code.
•
84
~th
The generator polynomial g{x) of the
code is given by g(x) = x
r
~(x) = (x
~(x-l), where
qt
1
- -l)/gD{x)
and r = N-k, k being the degree of gD{x),
root, ~(x) does not.
Since gD(x) has 1 as a
The roots of ~(x) are those elements
t
h
order Affine Geometry
.
a , 1 ~ h ~ q -2, such that Dq(pJh) 2 ~(q-l) for each j=O,l, ••• ,n-l.
Thus
~ (x
-1)
has as roots those elements a
t
.
1 < h < q -2 and D (pJh )
-
-
qt_ l _h
such that
> ~(q-l) for each j=O,l, ... ,n-l.
h' = qt_ l _h •
Then Dq (pjh') = t(q-l) - Dq-~
(pjh), and lL(x- l ) has as
h'
roots those elements orj=O,l, ••• ,n-l,
such that 1
~
h'
~
° < Dq (pjh') -< (t-~)(q-l).
t
q -2 and for each
This proves
Lemma 3.4.6 The generator polynomial g(x)
of the
~th
Geometry code of length qt_l has as roots those elements
1
~
h
~
Let
q-
t
q -2, such that
° < Dq (pjh) -< (t-~)(q-l)
order Affine
J1,
for each
j=o, 1, ... ,n-l.
Let hI = qt-Il....-1 = (q-l) + (q-l)q +
. +
( q-l)qt-~-l, "ll1ich we
may write as
+ (P_l)qt-~-l
hI = (p-l) + {p-l)q +
t-~-l]
+ ( p-l ) q
P
+[(p-l) + {p-l)q +
+ •••
+[(p-l) + (p-l)q + ••. + (P_l)qt-~-l] pn-l
From this representation, we see that for
° < Dq (pjh)
°< h
~
hI'
< D (pjh 1 ) = (t-~)(q-l), for each j=O,l, ..• ,n-l.
q
-
the roots of g{x) include the elements
2
Thus
h
a, or, ... , a 1 , and by the
BCH theorem, the minimum distance of the code is at least equal to
85
hI + 1 = qt-fJ.
This bound on the minimum distance of a fJ th order Affine
Geometry code will be strengthened in Section 3.6.
Likewise, the generator polynomial of the modified fJth order
Affine Geometry code has roots those elements
aP such that
o -< h -< qt_2 and Dq
(pjh) < (t-fJ)(q-l) for each j=O,l, .•. ,n-l. The
code is a subcode of the BCH code of length qt_ l specified by the
2
hI
t-p
roots l,ex,cr, •.. ,ex and has minimum distance at least hI+2=q
+l.
We now give a formula for the number of information symbols in
a vth order GRM code, where v = fJ(q-l) -1.
Recall that the function
R (t,q) was defined in Section 2.7 as
fJ
R (t, q)
(3.4.4)
fJ
=
~ Bs (t, q) ,
L
s=l
r
L (q)
s
where
Bs(t,q)
=
(_1)1
and where
s(q-l)
q
e~l) e+S(q~l)-iq)
J.
The function B (t,q) is the number of ways of choosing t+l
s
integers cO,cl, ••• ,c t such that 0
Co
+ ••• + c t
Theorem 3.4.2
=
~
ci
~
q-l, i=O,l, ..• ,t, and
s(q-l).
The number of information symbols of a vth order
GRM code of length qt_l with symbols from GF(q), when
'V
= fJ(q-l)-l,
is equal to K (t,q), where
fJ
(3.4.6)
K (t,q) = R (t,q) - R (t-l,q).
fJ
fJ
fJ
Proof.
From Definition 3.4.5, the number of information symbols
86
in a vth order GRM code is the number of integers h, 0 ~ h ~ qt_2,
< v.
such that D (h)
q
~(q-l)-l,
For v ::
this is equal to the number
of integers h, 0 ~ h ~ qt_2, such that D (h) < ~(q-l).
q
In the
proof of Theorem 2.8.6, it was shown that the number of such integers
h is equal to K (t, q).
~
Corollary 3.4.1
Hence the theorem is proved.
The number of information symbols of a
~th
order
Affine Geometry code of length qt_l is at least equal to K (t,q).
~
For q :: p, this is exact.
Proofl We have already shown that the (v ::
code is a subcode of the
~th
~(q-l)-l)th
order GRM
order Affine Geometry code and hence
the number of information symbols of the Affine Geometry code is at
least equal to that of the corresponding GRM code.
When q :: p, the
two codes are identical.
The properties of the Affine Geometrie codes discussed in this
section are listed in Table 3.2 for various values of the parameters.
For q
= 2,
these codes are identical to the cyclic Reed-
MUller codes and are listed in Table 3.3.
3.5 Non-primitive geometric codes
We turn now to a consideration of non-primitive geometric code&
We shall define two new classes of such codes.
Definition 3.5.1 Let a be a primitive element of GF(qt+l) and let
~:: a q- l • The Non-Primitive Generalized Reed-Muller ~.2! order ~
is defined as the cyclic code of length N :: (qt+l_l)/(q_l) with
symbols from GF(q) such that the generator polynomial of the dual
code has roots those elements
h
~
, 1
~
h
~
N-l, such that
87
o < Dq (h(q-l)) -<
~(q-l).
Definition 3.5.2
2
let ~
= a q- l •
Let a be a primitive element of GF(qt+l) and
The ~th order Projective Geometry ~ is the cyclic
code of length N
= (qt+l_ l )/(q_1)
with symbols from GF(q) such that
the generator polynomial of the dual code has as roots those elemenm
j
~h, 1 S h S N-l, such that 0 < D (p h(q-1)) < ~(q-l)
q
for some j
=
0,1, .•• , n-l.
-
(Here, q
=
pn).
The above specifications of the roots of the generator polynomials of the dual codes includes all roots, for if
o < Dq (pjh(q-l)) -< ~(q-l), then
o < Dq (pjqh(q-l))
=
D (pjh(q-l)) < ~(q-l), i.e., the conjugates of
q
-
each root ~h are also included in the specification of the roots.
We observe that the generator polynomial of the dual code of
the
~th
of the
order Non-Primitive GRM code divides that of the dual code
~th
order Projective Geometry code.
Non-Primitive GRM code is a subcode of the
Geometry code.
Hence the
~th
~th
order
order Projective
One of the mtivations for defining the
~th
order
Non-Primitive GRM codes is that the degree of the generator polynomial of the dual code may be easily computed; in fact, an exact
formula for it will be given later.
property of the
~th
However, the corresponding
order Projective Geometry code is more difficult
to deal with, in general.
Lemma 3-5.1
Both the
~th
order Non-Primitive GRM codes and the
~th
order Projective Geometry codes are non-primitive geometric codes of
This is an explicit formulation of the projective geometry codes
discussed by Rudolph [26].
2
88
order J..l.
Proof.
This lemma is immediate from Theorem 3.4.2.
The J..lth order Projective Geometry code is optimum in the
sense that, the number of information symbols of the code is the
maximum possible for a non-primitive geometric code of order J..l.
This follows from Theorem 3.4.2, for a necessary condition that a
code be a non-primitive geometric code of order J..l is that every
root of the generator polynomial of the dual code be also a root
of the generator polynomial of the dual code of the J..lth order Projective Geometry code.
Lemma 3.5.2.
We state tbis as a lemma..
Every non-primitive geometric code of order J1 and
length (qt+l_l)/(q_l) with symbols from GF(q) is a subcode of the
corresponding J..lth order Projective Geometry code.
Then h (q-l)=(q-l)+(q-l)q+••• +(q-l)q~
o
which we may write as
hO(q-l) = (p-l) + (p-l)q + ••• + (p_l)qJ..l
+ [(p-l) + (p-l)q + •.. + (P_l)qJ..l]p
+
+ [(p-l) + (p-l)q + ••. + (P_l)qJ1]pn-l.
From this, we observe that for each j = O,l, ..• ,n-l, Dq(pjhO(q-l»
(J1+l)(q-l), and for 1 -< h <- hO' 0 < Dq
(pjh(q-l»q
< D (pjhO(q-l»
.
(J..l+l)(q-l), i.e., 0 < Dq(pjh(q-l»
=
=
S J..l(q-l).
Let the generator polynomial of the dual code of the J,tth order
Projective Geometry code by ~(x).
2
~,~
, ••• ,
hO-l
~
The ~(x) has at least
h
as roots, but not
~
O
•
The dual code of the J..lth
order Projective Geometry code is thus a subcode of the non-primitive
BCH code of length N and designed distance hO' specified by the roots
2
~,13
hO-l
, ••• ,~
• The minimum distance of the dual code and the BCH
code is at least hOe
But the weight of the incidence vector of a
l1-flat in PG ( t, q) is equal to hO and by Lemma 3.5. 1, the dual of the
11th order Projective Geometry code contains at least +( t, 11, q) such
vectors.
This proves the following lemma.
Lemma 3.5.3
The non-primitive BCH code of length (qt+l_l)/(q_l)
with designed distance h = (ql1+ 1 _1 )/(q_l) and specified by the
O
2
hO-l
roots ~, ~ , •.. , ~
has minimum distance exactly hO' Moreover,
the code contains at least +(t,l1,q) vectors of minimum weight hO'
Furthermore, for 11~ p' ~ t-l, there are at least +(u,I1',q) code
'+1
vectors of weight (qP
-l)/(q-l)
The number of information symbols of the 11th order Non-Primitive GRM
code is equal to the degree of the generator polynomial of the dual
code, which in turn is equal to the number of integers h, 1
such that 0 < D (h(q-l»
q
~
h
~ N-~
< l1(q-l), where N = (qt+l_l)/(q_l). Since
-
h(q-l)50 mod q-l, then D (h(q-l»
5 0 mod q-l.
q
Recall from Section
2.7 that the number R (t,q) is the number of integers h, 1 ~ h ~ N-l
11
such that D (h(q-l» = s(q-l), 1 < s < 11. Hence, we have
q
Lemma
3.5.4
-
-
The number of inforlnation symbols in a 11th order Non-
Primitive GRM code of length (qt+l_l(/(q_l) is equal to R (t,q),where
I1
R (t,q) is defined by equation (3.4.4).
11
90
Since the
~th
order Non-Primitive GRM code is a subcode of the
~th
order Projective Geometry code, and for q = p, the two codes are
the same, we have
Lemma 3.5.5
The number of information symbols in a
~th
order
Projective Geometry code of length (qt+l_l)/(q_l) is at least
R (t,q); for q = p, this is the exact number of information symbols.
~
For the case
~
= t-l,
we give the exact number of information
symbols in a (t-l)th order Projective GeOm2try code as follows.
1
~
h
N-l, the element
~
~
-h
For
is not a root of the generator poly-
gD(x) of the dual code if and only if ~-h is not a root of
nomial
the incidence polynomial of some (t-l)-flat in PG(t,q), because, by
Definition 3.5.2 and Theorem 3.4.2, ~h' is a root of Sn(x) if and
h'
only if ~
is a root of the incidence polynomial of every (t-l)-flat
Also, ~O = 1 is not a root of ~(x).
in PG(t,q).
From Section 2.6,
p-h is not a root of gD(x) if and only if either h=O or else
h(q-l)
€
St' where 0 ~ h ~ N-l.
The number of integers h, 0 ~ h ~ N-~
such that ~-h is not a root of gD(x) is thus 1 + IStl, which is
1 +(
ptt-l) n
t
Lemma 3. 5. 6
3
.
We state this as a lemma.
The number of iuformtion symbols of a (t-l)th order
Projective Geometry code of length N=(qt+l_l)/(q_l), where q = pn,
is equal to
This result has also been obtained independently by Goethals and
Delsarte [10].
91
n
_ (P+t-l)
.
t
N - 1
Let us now consider the generator polynomial
g(x) of the
~th
order Projective Geometry code, given by
where ~(x) is of degree k = N-r and
~(x) = (xN_l)/~(X)
Since gD(x) has as roots those elements
° < D (pjh(q-l»
some j = O,l, •.. ,n-l,
j = O,l, ..• ,n-l, D (pjh(q-l»
q
Now
~
-h
=~
such that, for
< ~(q-l), 1 < h < N-l,
q
then g(x) has as roots those elements
h
~
~
-
-
-h such that, for each
> (~+l)(q-l), and the element 1.
-
N-h
and for h' = N-h, D (pjh'(q-l»
q
= (t+l)(q-l) - D (pjh(q-l».
q
So
g(x) has as roots those elements ~h such that
° -< Dq (pjh(q-l»
°< <
h
< (t-~)(q-l) for each j = 0,1,_ -,n-l, where
-
N-l.
Let
h
l
=(qy-~+l_l)/(q_l) •
Then hl(q-I) = (q-I) + (q-I)q + ••. + (q_I)qt-~, which we may write
as
hI(q-I)
= (p-I)
+ (p-l)q +
+ [(p-l) + (p-l)q +
+ (p_l)qt-~
+ (p-l)qt-~]p
+
+ [(p-I) + (p-I)q + . . + (p_I)qt-~]pn-l.
From this, we note that for each j = O,l, ••• ,n-I,
j
Dq(P hl(q_l»
= (t-~+I)(q-I) and that for
° ~ h < hI'
and
for each j
0,1, ... ,n-l,
° < D (pjh(q-l»
-
q
< Dq (pjhl(q-l»,
o -< Dq (pjh(q-l» -< (t-~)(q-l) for each j=O,l, .•• ,n-l.
i.e. ,
Thus g(x) contains as roots at least the elements
1,
2
t3, t3 ,
h -1
... ,
~ 1
,
hI
but not t3 • From the BCH theorem, the
~th
order Projective Geometry code has minimum distance at least h1+1.
We state this as a lemma.
Lemma 3.5.7
The minimum distance of the
~th
order Projective
Geometry code of length (qt+l_l)/(q_l) is at least equal to
(qt-P+l_l)/(q_l) +1.
Table 3.3 gives a listing of the parameters of some Projective
Geometry codes, including the lower bound on the number of information symbols from Lemma 3.5.5.
For t = 2, the first order Projective Geometry codes are the
Difference-Set Cyclic codes of Weldon [28].
;.6 Majority decoding of geometric codes
We describe here two methods of applying majority decoding to
geometric codes; one is a I-step majority decoding procedure and the
other, a
~-step
procedure.
First, we take the case of a non-primitive geometric code.
C be a
N
~-th
order non-primitive geometric code of length
= (qt+l -l)/(q-l).
Then the dual code CD contains the incidence
vectors of every ~-flat in PG(t,q).
Recall that the function +(t,d,q) is defined by
Let
.( t , d ,q ) =
(3.6.1)
t+l
t
t-d+l
isd+l
-l)(q -l) .•• (q
-1)
d
(q
-l)(q -l) ... (q-l)
By convention, .(t,-l,q) = 1.
The number of
~-flats
~-flats
in PG(t,q) is
.(t,~,q)
and the number of
which pass through, or contain, a given m-flat is
.(t-m-l,~-m-l,q). In
particular, the number of ~-flats which pass
through a given point is .(t-l,~-l,q) and the number of ~-flats
which pass through a given line or I-flat is
.(t-2,~-2,q).
Every vector of CD determines a parity check equation for C.
In particular, the incidence vectors of each
~-flat
in PG(t,q) de-
termine a parity check equation for C, viz., if g'=(hO,hl'.'.'~_l)
is the incidence vector of the
~-flat ~,
then, for every code vector
Let us label the points of PG(t,q) PO'IP "."PN- l such that
hi
=
{I, if Pi is incident with
~
0, otherwise,
where i
= O,l, •.. ,N-l.
Corresponding to the transmitted code vector
r' = (r '
o
... ,
r _ )
N l
£',
be the corresponding received vector and let
the error vector be e' = (eO,el,···,e N_l ), where r'
Let s
= h'r.
Then s
let
= h'r = h'c
since c ' is a code vector.
We
+ hIe
~ay
= c'
+ e'.
= hIe
write this as
We shall refer to equation (3.6.2) as a parity check on
eo' e1,··.,e _l corresponding to the
N
flat~.
Each of the
J
= .(t-l,~-l,q)
~-flats in PG(t,q) which passes through the point
Po determines a parity check equation which checks the symbol eO and
the coefficient of eO in each such equation is 1.
Suppose these J
equations are
Any two points, say Po and P., i > 1, determine a unique line,
l
say L.
-
The symbol e.l is checked by only those equations in (3.6.3)
corresponding to the ~-flamin PG(t,q) which pass through the line
L.
Since there are 5
= .(t-2,~-2,q)
such ~-flats, the symbol e. is
checked by exactly ~ of the equations (3.6.3).
l
From Theorem 3.).2,
the following decoding rule gives eO correctly, provided at most
[J/25] errors have occurred.
Decode eO as that value of GF(q) which
is assumed by the greatest fraction of the (s j) of equations (3.6 3) ;
in the event that no single value is assumed by a strict
pl~Tality
of the {Sj}' decode eO as zero.
The symbols
el,e2""'~_1
may be decoded successively by
choosing a different set of J equations which check e , etc.
l
determined the error vector,
~',
Having
the transmitted vector, .£', may be
recovered from the received vector, E,'.
Now J/5 is equal to (qt_l)/(q~_l), which, for large q, is
approximately equal to qt-~•
Theorem 3.6.1.
We state the above as a theorem.
Provided at most [(qt_l)/2(q~_1)] errors occur, a ~th
order non-primitive geometric code of length (qt+l_l)/(q_l) may be
correctly decoded using the one-step majority decoding algorithm as
described above.
95
Alternatively, it is possible to use a
procedure for a
~th
~-step
majority decoding
order non-primitive geometric code.
Two distinct ~-flats in PG(t,q) intersect in at most a (~-l)­
flat.
There are J l
(~-l)-flat,~.
=
.(t-~,O,q) ~-flats which pass through a given
Let the check sum of
E are Po,.··,Pk , then S
= eO
+ e1 +
~
be S,i.e., if the points on
+ ek .
check equations which correspond to the J l
in E.
Take the parity
~-flats
Which intersect
Each of these equations checks the sum S, but no other symbol
is checked by more than one of these equations.
Let the equations
be
(3.6.4)
This set of J
l
equations is orthogonal on the sum S.
By Theorem
3.3.1, the following decoding rule gives S correctly, provided that
at most [J /2] errors have occurred.
l
Decode S as that value of
GF(q) which is assumed by the greatest fraction of the {s.} in
J
equations (3.6.4); in the event that no such value is assumed by a
strict plurality of the {Sj}' decode S as O.
Likewise, the check sum of every (~-l)-flat in PG(t,q) may be
decoded using a majority rule.
These decoded sums may be treated
as new parity check equations, and the check sums of the (~-2)-flats
may be determined from these new equations by a majority procedure.
Since two distinct
there are J
2
(~-l)-flats
= .(t-~+l,O,q) >
J
intersect in at most a
l
(~-l)-flats
(~-2)-flat
and
which intersect in a
given (~-2)-flat, the check sum of each (~-2)-flat is given correctly by the majority procedure, provided at most [J /2] and hence at
l
most [J /2] errors have occurred.
2
96
This is repeated until, at the
~th
stage, the check sums of each
a-flat, i.e., each of the symbols ea,el, ... ,e _l , have been decoded.
N
Provided at most [J /2] errors have occurred, the check sum at each
l
stage is decoded correctly.
Theorem 3.6.2
We state this in the following theorem
Provided at most [J /2] errors have occurred, a
l
~th
order non-primitive geometric code of length (qt+l_l)/(q_l) may be
correctly decoded using the above
where
~-step
majority decoding procedure,
t-~+l 1
J
l
= ..9..-..:..-=q-l
We observe that for large q, J
so that the
~-step
l
is approximately equal to qt-~,
majority decoding procedure and the one-step
majority decoding algorithm of Theorem 3.6.1 will correct
the same number of errors.
approximate~
Also, the one-step decoding procedure is
easier and more economical to implement than the
Since the minimum distance of the
~th
~-step
procedure.
order Projective Geometry
code is, from Lemma 3.5.7, equal to J l + 1, the number of errors
correctable by the
~-step
majority decoding procedure is equal to that
guaranteed correctable by the minimum distance property of the code.
In Table 3.3, the number of errors correctable by one-step majority
decoding is given for these codes.
Except for small values of q, there
appears to be no appreciable advantage in using the
~-step
procedure
for the Projective Geometry codes.
Similar majority decoding procedures may be used for primitive
geometric codes.
Let C be a type a primitive geometric code of order
~
and length
qt -1., i. e., the dual code contains the incidence vectors of all
97
~-flats
t
As in the projective case, suppose the N=q -1
in EG(t,q).
points, other than the origin, in EG(t,q) are labelled PO,P l ,·· ,P - l ,
N
such that if hI = (hO,h l ,. "~-l) is the incidence vector of the
~-flat ~,
then, for i
hi
=
O,l, ..• ,N-l,
= {l,
if Pi is incident with
~,
0, otherwise.
The number of ~-flats in EG(t,q) which pass through a given m-flat
is '(t-m-l,~-m-l,q). In particular, there are '(t-l,~-l,q) ~-flats
Which pass through a given point and '(t-2,~-2,q) ~flats which pass
through a given line or l-flat.
Each of the J
= .(t-l,~-l,q)
~-flats
in EG(t,q) which passes
through a given point, say PO' determines a parity check equation
which checks the symbol eO'
For i
=
Suppose these J equations are
1,2, .•• ,N-l, the points Po and Pi determine a unique line.
Since there are ~ = .(t-2,~-2,q) ~-flats in EG(t,q) which pass through
this line, the symbol e.1 is checked by exactly
~
of these J equations
Hence, by Theorem 3.3.2, provided that at m0st
[J/2~]
errors have
occurred, eO is given correctly as that value of GF(q) which is assumed
by the greatest fraction of the {s.} in equations (3.6 5); in the case
J
no value is assumed by a strict plurality of the (Sj)' eO is given
correctly as O.
The remaining symbols el, •.• ,e _l are similarly decoded.
N
Analagous to the
case, a
~-step
~-step
decoding procedure for the projective
majority decoding procedure is available for the
primitive geometric codes of type 0 and order
~.
For a type 1 primitive geometric code of order
~,
the majority
decoding procedures are the same as the above, except that the only
flats used are those which do not pass through the origin.
There are qt-~ +(t-l,~-l,q) ~-flats in EG{t,q), of which
.(t-l,~-l,q) pass through the origin and (qt-~_l)+ (t-l,~-l,q) do not.
Let P be any point of EG{t,q), other than the origin.
~-flats
through P is
.(t-l,~-l,q).
Since P and the origin determine a
unique line, say L, and the number of
.(t-2,~-2,q),
~-flats
passing through L is
there are
J
~-flats
The number of
=
3
.(t-l,~-l,q) - .(t-2,~-2,q)
in EG{t,q) which pass through P but not the origin.
Suppose the parity check equations of all
~-flats
passing through
the point Po but not the origin are
(3.6.6)
For any i • 1, ••. , N-I, either the origin, Po and Pi are collinear or
they are not.
If they are, then none of these J
equations check both
3
the symbols eO and e i , for otherwise, the corresponding ~-flat would
pass through the unique line containing Po and Pi' which also contains
the origin, in this case; i.e., the
origin.
~-flat
would pass through the
If PO' Pi and the origin are not collinear, then the three
There are .(t-3,~-3,q) ~-flats in
points determine a unique plane, n.
EG{t,q) passing through..
There are
.(t-2,~-2,q) ~-flats
inEG{t,q)
passing through the line determined by Po and Pi' so there are
e3
= .( t-2, ~-2, q)
- +( t-3, ~-3, q)
~-flats
passing through Po and Pi but not the origin.
at most
e3
i
= 1,
of the J
••. , N-L
3
Consequently,
equations (3.6.6) check both eO and e i , for any
99
The one-step majority decoding rule of Theorem 3.3.2 gives eO
correctly, provided at most [J /25 ] errors have occurred.
3
The re-
3
maining symbols e ,e , ..• ,e _ may be decoded similarly.
l 2
Nl
The
~-step
majority decoding procedure for a type 1 primitive
geometric clde of order
~
is also similar to that for the non-primitive
geometric codes.
Two
~-flats
which do not pass through the origin either do not
intersect or else intersect in at most a
a
(~-l)-flat
and
~
(~-l)-flat.
Let
which does not pass through the origin
determine a unique
~-flat
~
Qe such
Then the origin
passing through the origin.
There
are .(t-~,O,q) ~-flats passing through ~, so there are
J
~-flats
4
=
+( t-~, 0, q) - 1
in EG(t,q) which pass through
~
but not the origin.
at most [J /2] errors have occurred, the check sum of
4
correctly by a majority rule from the corresponding J
equations.
Having decoded the check sums of each
~
Provided
is given
4 parity check
(~-l)-flat
which
does not pass through the origin, the procedure is repeated to give
the check sums of each
origin, etc.
At the
(~-2)-flat
~th
which does not pass through the
stage, the symbols eO,el, ..• ,e _1 are decoded
N
correctly, provided at most [J4/2] errors have occurred.
Since a
~th
order Affine Geometry code is a primitive geometric
code of type 1, we may apply the
described above.
~-step
majority decod.ing procedure
Since [J4/2] errors may be corrected, the minimum
( t-~+l 1)
distance of the code is at least J 4+l = +(t-~,O,q) =~ q-l .
100
TABLE 3.1
SUMMARY OF GEOMETRIC CODES
Length
Code
Roots of ~(x) are
PRDITTIVE GEOMEn'RIC CODES
J1
vth order Reed-Muller
D2 (h) ::: v
vth order GRM
t
q -1
vth order EG
2nt _l
f.,lth order Affine Geometry
t
q -1
such that
D (h) < v
q
-
?
for some j=O, .• n-l
~h such that
NON-PRIMITIVE GEOMETRIC CODES
vth order Modified Reed-Muller
(2 t +l _l)/(2_1)
f.,lth order Non-Primitive GRM
(q
t+l
-l)/(q-l)
0 < D (h) ::: v
2
a < Dq (h(q-l))
::: f.,l(q-l)
f.,lth order Projective Geometry
t+l
(q
-l)/(q-l)
a < Dq (pjh(q-l))
::: f.,l(q-l)
Difference-Set
(~ -l)/(q-l)
D (pjh(q-l)) = (q-l).
q
101
TABLE 3.2
PARAMETERS OF AFFINE GEOMETRY CODES
q
4
t
2
3
4
5
8
2
3
3
2
3
4
I.l
1
1
2
1
2
3
1
2
3
4
1
1
2
1
1
2
1
9
5
2
2
3
Info. Symbols
> K (t, q)
-
I.l
15
63
63
6
10
44
255
255
255
1023
1023
1023
1023
15
106
221
21
222
667
968
28
84
63
511
3
1
2
511
8
26
26
80
80
80
242
242
3
4
242
242
1
1
1
2
80
2
5
Length
t
N=q -1
24
124
124
392
3
4
17
5
31
66
6
51
147
222
36
10
20
10
Minimum
Errors Corrected by
Distance
1-step Decoding
[J /25 ]
(qt-l-l_ l )/(q_1)
3 3
5
21
5
85
21
5
341
85
21
5
9
73
9
4
13
4
40
13
4
121
40
13
4
2
10
2
42
8
2
170
36
8
2
4
36
4
1
6
1
19
4
1
60
15
4
1
4
10
6
2
31
6
15
2
102
TABLE 3.3
PARAMETERS OF PROJECTIVE GEOMETRY CODES
q
t
~
2* 2
1
3
1
4
2
1
2
5
6
3
1
2
3
4
1
2
3
4
5
7
1
Info. Symbols
> R (t,q)
-
J1
7
15
15
3
4
10
31
5
15
25
6
21
41
31
31
63
63
63
63
127
127
127
127
127
56
7
28
63
98
119
8
255
255
255
36
218
246
1
255
255
255
511
2
511
3
4
6
511
511
511
511
255
381
465
7
511
501
2
3
4
5
6
8
Length
qt+l - 1
q-l
5
92
162
9
45
129
Mininrum
Distance
qt-~+l - 1 +1
q-l
4
8
4
16
8
4
32
16
8
4
64
32
16
8
4
128
64
32
16
8
4
256
128
64
32
16
8
4
Errors Corrected by
I-step Decoding
(qt_l)/(q~_l)]
[l
1
3
1
7
2
1
15
5
2
1
31
10
4
2
1
63
21
9
4
2
1
127
42
18
8
4
2
1
* For q=2, the projective geometry codes are the cyclic Reed-MUller
codes.
103
TABLE 3.3 (contd)
q
t
1.1
Length
Info. Symbols
4
2
3
1
1
2
1
2
3
1
2
3
4
1
1
2
1
2
3
1
1
2
1
1
1
2
1
2
3
1
2
21
85
85
341
341
341
1365
1365
1725
1365
10
20
64
35
170
305
56
392
972
1308
36
120
464
330
2340
4350
136
816
3552
528
6
10
29
15
60
105
21
111
252
342
45
4
5
8
2
3
4
16 2
3
32 2
3 2
3
4
5
3
9
2
4
1
73
585
585
4681
4681
4681
273
4369
4369
1057
13
40
40
121
121
121
364
364
364
364
91
Distance
6
22
6
86
22
6
342
86
22
6
10
74
10
586
74
10
18
274
18
34
5
14
5
41
14
5
122
41
14
5
11
I-step errors
2
10
2
42
8
2
170
34
8
2
4
36
4
292
32
4
8
136
8
16
2
6
1
20
5
1
60
15
4
1
5
104
TABLE 3.3 (contd)
Info. Symbols
Distance
q
t
I.t
Length
9
3
1
2
1
1
1
2
820
820
165
654
92
11
45
4
757
31
156
156
378
15
29
7
32
14
27 2
5 2
3
35
120
7
l-step errors
3
15
2
105
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