ON THE CONSTRUCTION OF BOSE ClIAUDHURI MATRICES
WITH '!HE HELP OF ABill..IAN GROUP CHARACTERS
by
DOMINIQUE C. FOA.TA
Insti tute or Statistics
Mimeograph Series No. 278
March, 1961
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. c.
AFOSR Report No.
489
ON THE CONSTRUCTION OF BOSE-CHAUDHURI MATRICES
WITH THE HELP OF ABELIAN GROUP CHARACTERS
by
Dominique C. Foata
University of North Carolina
March, 1961
Contract AF 49(638)-213
It is shown how matrices used in
error-correcting codes can be derived from Abelian group characters.
Qualified requestors may obtain copies of this report from the ASTIA Document Service Center, Arlington Hall Station, Arlington 12, Virginia.
Department of Defense contractors must be established for ASTIA services
or have their "need-to-know" certified by the cognizant military agency
of their project or contract.
Institute of Statistics
Mimeograph Series No. 278
ON THE CONSTRUCTION OF BOSE-CHAUDHURI ~~TRICES
WITH THE HELP OF ABEUAN GROUP CHARACTERS *
by
Dominique C. Foata
University of North Carolina
Bose ~L7 has shown that the existence of an n x r matrix A
...7
ith entries from GF(s) (s prime pOi'ier) having the Pd property that
any d
ro~s
of A are independent, was equivalent to the existence of
an (n, n-p) s-ary t-error correcting and (t + l)-error-detecting group
code if d = 2t + 1.
He also proved that for d = 2t, it was equivalent
to the existence of a
1
Sn fractionally replicated factorial
Sn-p
design in which no t-factor or lower order interaction was aliased
with any t-factor or lower order interaction.
Thus we can build error-correcting codes or fractionally replicated
factorial designs as soon as we have constructed such matrices having
the Pd -property.
Bose and Ray-Chaudhuri ~!.7 and ~'[] have gtven
an explicit method of construction in the binary case.
Peterson ~~7
has investigated some properties of the codes built from these matrices.
In particular he gave the exact value of the ranks of these matrices.
Finally Zier1er~~7 has generalized these results to the s-ary case
(s prime power).
In this paper using the theory of group characters we reformulate
these results and show how these matrices can be obtained from the
*This
research was supported in part by the United States Air
Force Office of Scientific Research of the Air Research and Development
Command, under Contract No. AF(638)-2l3. Reproduction in whole or in
part is permitted for any purpose of the United States Government.
2
character tables of cyclic groups.
Hence considering character
tables of Abelian groups we can obtain an analogous construction
and so a new family of matrices having the pd"property.
1.
In this section we define the Pd-property over characters of
an Abelian group and investigate some properties involved by this
definition.
Let G be an Abelian group of order g whose invariants are:
h l , h , ••• , h
2
r
We know that these invariants are characterized by the following
properties:
(i) G is a direct sum of r cyclic groups G , G2 , ••• , G of
l
r
... , h
(ii)
hi+l / hi
r
(i
respectively
= 1,2, ••• ,(r-l)
Thus every element a of G may be uniquely represented in the form:
a
..• c
r
r
Later on we shall speak about the r "coordinates" of a.
Furthermore if 11 is a field whose characteristic does not divide
h=hl and if ~i is an hi th primitive root of unity in
the characters of the group G are given by:
ula l
- - . . , ) ~l
a
€
n
(i=1,2, ••• ,r),
u a
u2a 2
~2
r r
• •• ~r
G
(See Van der Waerden page 175 vol. 2)
We will also denote the character X(u ,u , ••• ,u . ) by
l 2
r
~1 ul~2 u2 •.. ~ ur
As hi divides h for all i=2, ••• , r, we can take Si as a
well-defined power of
Sl=~
and thus all the images of the elements
of G under all the characters will be powers of S.
In writing down the character table of G, let us make each row
correspond to one element of the group G and each column to one
character, in a 1-1 manner.
At the intersection of the row a (a
€
G)
and the column X (X a charac ter of G) we write the image of a under X,
i.e. X.(a).
Hence the above statement simply says that all the entries
of the character table of G are powers of
~.
Now let p be a prime number not dividing h and let p have the
order m in the residue system modulo h (pm;; 1 (h) and pm~1 11 (h)
t
for m
m 1
< m). Furthermore let c = E ~
of the Galois field GF(pm).
unity in GF(pm).
s,
and n be a primitive root
Then .c is an h ~ primitive root of
Hence if we take GF(pm) for the field{l and x c for
all the entries of the character table of G will be elements of the
Galo:is' field GF(pm) and all the properties rin the group characters
will be preserved.
In particular, as the character table of an
Abelian group of order g is a non-singular matrix of order g, we have:
Proposition (1.1) Given an Abelian group G of order g whose invariants
are (h, h2 , ••• , h) and a prime p not dividing h, we can construct
a non-singular matrix of order g whose entries are from
GFlt?), m
being the order of p in the residue system modulo h.
In the following it is assumed that the prime p has been
definitely chosen (prime to the order g of the group) and the
characters take their values in the well-defined field GF(pm), m
being fixed by the choice of p.
Moreover the character table or
the group of the characters of G will be designated by
simply
~ .•
~(G,p,m)
or
4
Let us recall the definition of the Pd property introduced
by R. C. Bose 1:"1:...7:
Definition 1. A matrix whose entries are from a field
fL ,
has the Pd property if any d rows of this matrix are linearly
independent
We shall also say:
Definition 2.
A set of
e characters
(Xl X
2
~ (G, p, m) has the P property (over GF(pm»
d
character table
~I!
••• X ) of the group
e
if the submatrix of the
formed by the r1 columns Xl' X 2 '
... , Xe
is such
that any d rows are linearly independent.
Before introducing the definition of the Pd property over a
subfield, let us define an equivalence relation among the characters:
It is known that to each diVisor n of m corresponds a subfield
GF(pn) of GF(pm); and the Galois group of GF(pm) over GF(pn) is the
cyclic group
mof
order m
l
=m
n-
generated by:
q = p
n
Hence the definition:
Definition 3.
Two charaClters Xl and X 2 of ~ (G,p,m) are
equivalent modulo (}. ,(n) if there exists en.-integer' e such that:
e
X = X q
(q' = pn)
1
2
As ~ is cyclic and
~vidently
m
ml
xP = xq = X'for
all~, this relation is
an equivalence relation.
* X2*'
We denote the equivalence classes modulo ~ (n) by Xl'
and the set of characters in the class
,~
by L~
*1
; •
~
*
"containing the character X
Definition 4.
(Pd property over a subfield)
A set of echaracters (X I X •••X ) of
2
e
~
5
(G,p,m) has the Pd property
over GF(pn) (n divisor of m) if any d row vectors of the submatrix
(X X ••• X ) of ~. are linearly independent over GF(pn), i.e. if
I 2
e
v. , v. , ••• ,v
1.
1"
1.
2
id
are d row vectors of the submatrix (Xl X •••X ),
2
e
a relation of the form,
+ Ad v i
for
~,A2' ••• , Ad
€
::;; 0 cannot hold
d
subfield GF(pn).
We show:
If (Xl ,X2 , ••• ,Xe ) of ~(G,p,m) has the Pd property over
GF(pm), then (Xl* X* •••X* ) has the Pd property over GF(pn ).
2
e
Proposition(1.2)
It is sufficient to show that if
~ X(a ) + ~ X(a ) + ••• + Ad X(ad ) ::;; 0
k
2
Al ~ ••• Ad
€
a l , a 2 , ••• , ad
X
€
L1
GF (pn )
G
€
(G,p,m)
then the same linear relation holds if we replace X by any other
character of the equivalence class to which X belongs.
n
Indeed as a
------> a P
is an antomorphism of GF(pm) leaving the
elements of GF(pn) invariant elementwise, we have:
q3
q (ml-l)
q2
Thus the same relation holds for Xq and hence for X , X , ••• ,X
,
that is, for all the characters of the class X* containing X.
Conversely:
Proposition (1.3)
If the set of classes (X~, X;, ••• , X*
e ) has the Pd property
6
over GF(pn), then
(\X~\,{ x;l
, ... ,{
X:~)
Suppose there exist· d elements of G,
d
Z "l\ X(a )
i=l i
i
has the Pd-property GF(pm).
al~a2,
=0
••• ,ad such that
(i=l, •.• ,d)
and this relation holds for all the characters X from
({X~ J'{ X;] ~ ...,{~: J).
Then as a ~ a
d
Z "l\.x(a )
i
i=l ~
q
(q=pn) is an automorphism of GF(pm),
d
d
2 q2
d q (ml-l) q (ml-l)
x (a )= 0
xq(a ) = Z"l\~ x (a.)= •.• = Z "l\i
i
i=l ~
i
1=1 ~
~
i=l
= 0 ~ Z"l\~
Hence:
since the above relation is assumed to hold for all the characters of
of the same equivalence class.
But on the other hand ~.= ("l\ + "l\q+
~
GF(pn) since
~~=~.
~
i
i'"
(m -1)
+"l\q 1
) is an element of
i
(i=l •.. d).
~
Hence we have found a linear relation over GF(pn) between d
elements of G:
d
Z
i=l
~i X(a.)
=0
i=l, •.• ,d which
l.
holds for all the elements of ({ X~
J ,... , {X:}
) 1. e., for the classes
* ••. ,X* ) themselves and this contradicts our hypothesis.
(Xl'
e
(ml-l)
If it happened that "l\.+ "l\iq+... +"l\~
were null for all i=l, ... ,d,
~
~
d
we would multiply the equality Z "l\.x(a ) = 0 by a suitable element v
i
i=l ~
of GF(pm) such that the relation:
7
would not hold for all the
This is always possible since
~i's.
the equation:
~-l
~ + ~q + ... + ~q
=0
is not satisfied by all the (qm_l )
non null elements of GF(pm).
Thus we can say:
Proposition 1.4.
The set (Xl'¥' ,X2* "",Xe * ) of e distinct classes
modulo ~ (n) of L (G,p,m ) has the Pd property over GF(pn)
only if, the set
({xl *} , tx2*}, ..., {xe1)
1
if and,
has the Pd property
over GF(pm) •
In particular, if we take all the classes modulo ~(n) , we
exhaust all the characters of the group; and since the character table
is non-singular or has the P property over GF(pm) , we have:
g
Proposition 1.5.
the P
-g
The set of all the classes modulo ~(n) has
property over GF(pn) •
h
As X
=1
for all X
€
L (G,p,m) , the inverse of X is given by:
X = xh - l
Hence, if Xl
Xl =
i.e.
x2Qk,
B
X modulo ~(n)
2
taking the (h-l)
S
t
power of each member, we obtain:
"
) (X h-l)q
X (
h-l
1
=
2
k
-
or Xl
= X- 2q
k
-* •
Thus we can speak of the inverse of a class X* , we shall denot by X
*
*
*
if (Xl ' X2 ' ... , Xe ) has the Pd property over.
~ *' X
- *' ••• , X- *) has the P property over GF(pn).
GF(pn) , then (A
d
e
l
2
Propolition 1.6.
d
For, if
L
i=l
then
d
L
i=l
~.
X(ai) = 0
::L
~i X(81 1 = 0
l
, since X (a- )
= X(a)
8
Hence each linear relation between d row vectors of the submatrix
(Xl ,X2 ,c •. ,X e ) where Xl
€
Xl* ' ••• 'Xe
Xe*
€
(Xl ,X2 , ••• ,Xe ),
between d row vectors of the sUbmatrix
... , X
e
Notation:
X
€
e
implies a linear relation
where
*
We designate by n
i
*
the number of distinct characters
contained in the class Xi* •
Definition
4: n being gixed as divisor of m, we say that a
character X of E(G,p,m) belongs to the Galois field GF(pe) if e is
the least multiple of n such that GF(pe} contains all the images X(a)
(a€G) •
Proposition
1.7. If X belongs to GF(pe}, the number n* of characters
of the class X* is equal to: n*
= e1 = ~n
For if X belongs to GF(pe}, all the characters contained in X* also
belong to GF(pe).
~-l
X(a),X(a)q, ••• ,X(a)q
can only have e
Hence the class X* only contains n*
distinct elements.
l
= el
distinct characters.
Finally we remark:
Proposition
1.8. If a set of characters of E(G,m,p),
~Xl'X2, ••• ,Xe)
has the Pd property, then the set (XX 'XX2 ' ••• ,XXe ) has the Pd property.
1
(We denote by XXi the character: a --;;. X(a) Xi (a) )
d
For i f there is a relation: E
i=l
for
j = 1 ••. e,
where A.
i
€
GF (pm) i
\
X(a.) X.(a ) = 0
J
1
= l ...d
i .
•
= lJ. is an element of GFU/!l).
Then, "J...1 X(a.)
1
i
Hence there exists a relation:
d
E
lJ. X. (ai )=0
j = 1 ... e
i=l i J
which contradicts the hypothesis of the Pd property.
9
Example 1:
Consider the cyclic group G of order g = 15.
15
Choose p = 2, (2, 15) = 1, and 2 has the order 4 in the residue system
modulo 15
Hence~
being a primitive root of GF(2 4 ), in fact a 15th root of
unity in this field, we can take our field(;, into which the characters
take their values, as the Galois field GF (2 4 )
, 2, 4) are so:
15
ca ~ x ua
a=0,1, •• 0,14
The characters of E(G
u
t
U
= 0,1, ••• ,14
To n
=1
corresponds the equivalence relation
~(l)
and the
following equivalence classes:
X*
0
=1
1
*
~,
Xl
X*
2
~2, t 4 ,
t8
t 3, ~6, ~12, ~9
= ')(2*
- *
X* =X
3
3
~5, t lO
')( *
~7, ~14, ~13, ~ll
1
*
*
*
* - *
implies that (X o ' Xl ' X2 ' X3 ' Xl ) or simply one
representant of each class (1, ~, t 3, t 5 , ~7) has the P property
15
over GF(2) •
(1.5)
The inverses of X2* and X * are X2* and X *
3
3
As proved in R. C. Bose (1), the set (~, ~3, ~5) has the P6
property over GF(2).
Hence by 1.6 the set
-
(~, t 3, ~5) has also the
P10 property over GF(2), i.e. (~7,t3,~5)
The character ~5
ca ~ x5a
Hence the values taken by t 5
are x5 , xlOand x15
= 1.
Thus ~5 belongs to GF(22)
and the equivalence
lO
class to which it belongs, only contains two characters ~5 and t •
10
ExamE~~_~
Consider the Abelian group G) direct sum of the cyclic
group of order
G
15
and of the cyclic group of order 3:
= G15 (B G3
4
We have just seen that for G we can take GF(2 ) for the field;l ;
15
4
th
x, the primitive root of GF(2 ) is an 15 primitive root of
Hence x5 is an 3 rd primitive root of unity.
unity.
Thus the character of ~ (G,2,4) are:
ua l 5va2
(al~a2) ~ x
x
u
= 0,
1, .•. , 14 a
v
= 0,
1, 2
= 0,1, ..• ,14
1
= 0,1,2
a2
To the equivalence relation
~(1)
correspond the classes:
1
y 2
~ 4 ~18
~1' ~1 ' ~1 ' ~
p
~2' ~2
2
.3
6
12
9
~1 ' sl ,sl ' sl
~1~2'
S1
2
~22, ~14~2'
2
8
S1 ,S2
24282
Sl S2' ~l S2 ' ~1 ~2' ~1~2
~13S2' S16~22, ~112~2' S19~22
~
~1
5
~
'~1
10
3 2 . 6
12 2
9
~1 ~2 ;~1 s2' sl s2 ,sl s2
5
10 2
Sl ~2'~1 S2
~
-*
Xl
X*
9
X*
4
*5
X
~1
-*
=X
9
7 ~ 14 ~ 13 ~ 11
'~1
'~1
'~1
S15~22, ~110~2
7
14
13
11 2
~1 '~2' ~l 'Sl S2'Sl ~2
Sl
7
2
14
13 2
11
S2' Sl ~2' ~l S2' Sl ~2
•
II
These 14 classes have the P property over GF(2).
45
2
345
2
2 2
Later we shall prove that (l,tl'~l ,t l ,t l ,t l ,t2'~2 ,t l t2 ,tl t 2 'Sl t~
4
has the P6 property over GF(2 ).
Hence the set of classes (X o* ,Xl* 'X2* 'X * ,X 4* 'X * 'X * ) has the P6
3
5 7
property over GF(2).
Hence by (1.6) the set of the inverse classes
- *'X2* 'X
(Xo* 'Xl
*'X *'X *'X * ) has the P6 property over GF(2) or if we
3 4 5 7
pick one element from each of these classes:
the set (l,t17,t2,t13,t17~2,t17t22'~15)has the P6 property over GF(2).
-2-
If a character X of E (G,m,p) belongs to an intermediate field
GF(pe») (GF(pn) c GF(pe) c GF(pm) ), we will show, in this section, that
the images X(a)
e =e
length n
l
(a~G)
= n*
can be represented isomorphically in a vector of
(the number class of X) with coordinates from GF(pn ).
We will call this vector P(X(a), n* ).
Hence assuming that (Xl ,X2 , ••• ,Xe ) is a set of non-equivalent
characters of E(G,m,p) having the Pd property over GF(pn), the substitution
X(a) ~ P(X(a), n* ) in the submatrix (Xl'." ,X e ) will yield to a matrix,
with entries from GF(pn), having the Pd property.
We shall use the following theorem~
From C. C. Mac Duffee "An introduction to abstract algebra" page 109:
"L e t f( n ) = xn + alxn-l + ••• + a
x + a b e a po 1ynomia1 wit h
n-l
n
coefficients in a field F and irreducible over F.
Let p be a root of this polynomial and consider the matrix:
12
•
o
=
R
0
1 0
o1
0 - a
·..
·..
0 - a
0 - a
n
n-l
n-2
1
0 0
·.. 1
- al
Then the correspondence:
n l
1 pn-l ~ c I + clR + ••• + c
R - =A
n0
n-l
is biunique and is an isomorphism under both addition and multiplicat ion. II
a
= co+c l
P + ••• +
C
Denote the latter field by K.
Hence:
Prowsltion 2.1. A set of k matrices from K are linearly dependent
over F, if and only if, the first rows of these k matrices are linearly
..>
dependent over F.
"Onl y" is trivial.
If the first rows of k matrices A A , '0 .,Ak of K
1 2
are linearly dependent, then there exist k elements of
such that
~l
But B
€
+
~2A2
+ ••• +
~Ak
Fr~'~2""J~
is a matrix B whose first row is null.
K and admits an inverse unless it is null.
As h&ving its
first row null, B is singular and therefore is the null matriX.
FroJtqsition 2 .2.
We can express this by saying:
A set of k elaments
a l , cx2 ' ••• , a k of F(p) are linearly independent over F, if and ~nly if,
the first rows of the corresponding matrices of K, A , A2 ,
l
the isomorphism a
<
> A are
~O'1 ~k
in
linearly independent.
Let us apply this result to our group characters from E(G,p,m):
If n divides m, the field GF(pm) is an algebraic ext~Dsion of GF(pD)
of degree
~
=m
-n
Thus every element of GF(pm) can be iSO~Lr'1'phically
expressed as a matrix of order ~ with entries from GF(pn).
13
In our character table
~(G,p,m),
if a character X belongs to
an intermediate field GF(pe), all the images X(a) (a
€
G) can be
expressed as matrices of order el = ~ = n* with entries from GF(pn),
under the above isomorphism we shall denote:
X(a)
~
M (X (a), n* )
(The order of these matrices M(X(a), n* ) is equal to the degree of the
extension of GF(pe ) over GF(pn ); that is, to n* , the number of characters
in the class X* ).
The first rows of these matrices M(X(a), n* ) will be denoted by
P(X(a), n* ).
(2.2) implies:
Proposition 2.3.
If a set (Xl\X2' ••• '~) of non-equivalent characters
of E(G,p,m) has the Pd property over GF(pn) (n divisor of m), then we
can construct a matrix of g rows and (n*l + n* + ••• + n* ) columns with
2
k
entries from GF(pn) which has the f d property.
*
*).
Moreover the rank of this matrix is (n*
l + n2 + ••• + nk ,
n*1 (i
=1
••• k) being the number of characters in Xi* •
Indeed from (2.2) we deduce:
Xi(a ), x i (a2 ), ••• , xi(ae ) are linearly independent over GF(pn), if
l
*
and only if, the e vectors of length n.:
~
p(xi(a l ), n*i ), P(X i (a2 ), n*i ), ••• , p(xi(ae ), n*i ) are linearly independent.
(i
=1
••• k)
Hence if we replace each element, xi(a), in the sUbmatrix (Xl' X2 , ••• ,Xk )
of E (G,m,p), by P(Xi(a), n* ), we obtain a matrix with (n*l + n*2 + •••+n*k )
i
columns and g rows and the Pd property is preserved.
Furthermore this matrix is of rank (n* + n* + ••• + n*k ). For if
2
l
* X*2 ' ••• , Xg*)
* and pick one
we take all the character classes (Xl'
representant from each class (Xl' X2 , ••• , Xg*), by (1.5) the matrix
(Xl' X2 ' ••• , Xg*) has the P property over GF(pn).
g
14
•
*
Hence if we replace each element Xi(a) of this matrix by P(X.(a),n.),
1
1
we will get a non-singular square matrix of order g, since
*g
••• +n*=g.
This. implies that the rank of the matrix (Xl'X2' •• o,~) after
having made the substitution Xi(a)
~
*
P(Xi(a), n*i ), is (n*
l + ••• + nk ),
which is the number of the columns.
-3We can now apply these results to cyclic groups.
We so obtain a reformulation of the results of Bose and Ray-Chaudhuri
~17 and ~g7
in the general case.
The result of Peterson ~~7 on the
number of columns of the Bose-Chaudhuri matrices having the P property
2t
over GF(2) is also being generalized and the present proposition follows:
Proposition 3.1.
Let G be a cyclic group of order h, p a prime
number not dividing h, m the order of p in the residue system modulo h
and n a divisor of m.
Then for d given, we can construct a matrix of h rows and R(h,d,n)
columns with entries from GF(pn) having the P property.
d
R(h,d,n) is given by the number of residue systems mod h among the
integers:
qju (u
= 1,2, ••• ,d
j
j>o) q
= pn
th primitive root of unity in
Under these assumptions x c is an h
m_
= ~l and x is a primitive root of GF(pm).
Hence the group E(Gh,p,m) consists of
X
u
= ~u:
b a ~ x ca
u=O,l, ••• ,(h-l) a=O,l, ••• ,(h-l)
The submatrix
(X l ,X2 ,.o.,Xd ) of E(Gh,p,m) has the Pd property since
any set of d rows bal, ba2 , ••• , bad yields to a Bquare matrix:
15
cal
Yo
ea
x
2
2ea
l
x
deal
x
2ca
dca
2 ... x
x
2
...
x
cad
x
dca
2cad
x
d
which is non-singular, since its determinant is a Vandermonde determinant.
* X.* , ••• , Xi* *) obtained from (X , X , ••• , Xd )
Hence the set (X.,
1
12
d
1
2
by retaining only one representant of each class has the Pd property
But the congruence:
X. ;: X. modulo
J
1
means that
i
2
j
j !!
~(n)
q~h) for a certain k
,
l
i qk(h) for a certain k
•
Hence the two sets iqu
and
u = 0,1, ••• ,m -1
1
j qV v = O,l, •• o,~-l
are the same.
*1 *1 ••• 'X*i ) is
Thus the numder d* of the class characters (x"X"
1 2
.d.*
equal to the number of distinct sets among the d sets: (jqU;~o) j=1,2, ••• ,d
these numbers taken modulo h.
* ~ p(Xi(a);n
*
*i )
Now if we make the substitution Xi(a)
(i = i l ,i 2 , ••• ,i d*), a
P
€
G , in the submatrix (X.* ,X.* ""/X,*
h
11
12
1d*
), the
d property is preserved by (2.3) and the number of columns we obtain·'"
is equal to: nr + nr +••. + n~ , the number of different residue systems
1
2
ld*
modulo h among:
qju
(u = l,2, ••• ,d; j ~ 0)
since by (1.7), n* is
i
equal to the number of different distinct residue systems mod h among
i qU
u
= 0, 1, •.. ,illl -
1
16
4.
In this s<Jction ~1~ shall pr~s,mt., 'J.p to d
= 6,
sets of
characters of an Abelian group which have the Pd-property.
Hence
using the techniques of section 1 and 2, we can construct matrices
having the Pd-property with entries from some Galois field of characteristic p, when p does not divide the order of the group.
Let G by an Abelian group whose invariants are (h ,h , ... ,h r )
l 2
u
u
u
l
2
and let ~l ~2 • . . S r (0 < u < h - 1,
< u2 -:: h2 - 1, ..• ,0 < u <: h -1)
1- l
r
r- r
°
• h
r
characters.
We have just seen that when r
=1
(cyclic group},the set
(t,~2, ... ,~d) had the Pd-property.
What can we say when the number of
We shall use the same method:
inv~riants
is greater than l?
in order to prove that the set
(X l ,X 2 "" ,X~,) of characters of G has the Pd-property, we shall show
that in the submatrix of I: formed by the e columns (X. ,X. , •.• ,x.~.)
l 2
and any d rows, there always exists a square matrix of order d, which is
non-s1ngulnr.
u
u
.
/" 1"~2'
2 . .
We mak e t h e conventlon
that any c h aracter ~l
in which an exponent u
i
is greater than hi' vanishes.
shall denote by Pi the function: a ~ a
of G, a
1
= Pi(a)
u
I'
~r
r '
Moreover, we
(a being an element
= (a l ,a2 , ••• ,ar ).
(4.1)
Consider two
G.
They differ at
Then the submatrix
which is non-singular.
17
(4.2)
~2'~22""'Sp'~p2) has
The set (Sl'S12,
Again if the
coordinates of a
i~
" ,. 2)
sub matrix ( Si'~i
is:
(
~1
and a
ali ,/ali )
a
~i
2i
Si
l
2a
2i
2
the P2-property.
are different, the
which is non-singular.
Consider three elements of G:
If there exists a coordinate i, in which they all differ, we pick the
2
subset (l'Si'Si ) and the corresponding submatrix is:
2a
S ali
li
1
Si
i
2a ,
S a2i
S 2~
which is non-singular.
1
i
i
2a
S a3i
3i
1
Si
i
If it does not happen, there exists, however, a coordinate i in which
8.
differ.
1
differ and also a coordinate k ~ i in which a and a
l
2
3
We then choose the subset (l'~i'Sk) and the corresponding suband a
matrix is:
1
ali
Si
1
s62i
1
s~li
Its determinant is:
i
~
a
+(s 3 k __
-
k
null since a2i~ ali and a3k~ alk'
a
lk
Sk
a
2k
~k
a
3k
Sk
~ ka lk ) ( !; ia 2i _ ~~li)
~
which is non-
18
Proposition 4.4
p
~2 r3
r r
Th e se t ( l'~i'~i'~i; ~i~j; i=1,2, ... ,r and
property.
Let a a a a be 4 distinct elements of G.
l 2 3 4
We have 4 cases to
consider:
a)
The ~th coordinates alk,a2k,a3k,a4k' of these four elements
are distinct.
But then from the set
1
I
alk
a
a
2k
3k
84k
1
(;
k
a
(; lk
k
a
2k
1
~k
1
a
3k
(;k
1
a
" 4k
Sk
(l'~k'~~'S~) we
~2
k
2a
~k
2a
~k
lk
2k
2a
3k
~k
~2a4k
k
~3
k
3a
~k
obtain matrix:
lk
3a 2k
~k
3a
~k
3k
3a4k
~k
which is non-singular (Vandermonde matrix).
b)
a
There exists a coordinate k for which three elements, (say)
a , have their coordinates distinct and the fourth one a4 has its
2 3
~th coordinate a4k equal to 8 (say). Then there exists another coor3k
dinate i such that P (a ) = a
~ a = P (a ).
i 3
4i
i
4
3i
l
a
In this case, the subset
1
*
*
I 1
(l'~k'(;~'(;i) gives the submatrix:
p
~k
alk
~k
1
a
2k
Sk
a
3k
1
a
3k
Sk
1
~k
(;2
k
2a lk
~k
2a
2k
~k
2a
3k
Sk
2a
3k
(;k
*
*
a
S 3i
i
a .
S 4J.
i
19
and the determinant of this matrix is:
a
1
~ lk
k
a
c)
In
2k
1
~k
1
a
3k
Sk
~ 0
, k, 3 coordinates are equal and the fourth one is
different, or:
Pk(a )
l
= Pk (a2 ) = Pk (a3 ) = a lk
and
Then there exists another coordinate i with Pi(a )
l
b) •
= ali~
a
2i
= Pi (a2 ).
If ali' a 2i and a 4i = P (a4 ) are distinct, then we are in the case
i
Hence suppose 8.4i is equal to ali (al and a 2 play the same role).
= Pi (a3 ) is different from ali or a 2i we are again in the
3i
case b). Thus we are left with: P (a ) = ali and then there exists
i 3
another coordinate j for which P (a ) = a
~ a
= Pj(al ) or
j 3
lj
3j
Now if a
P (a )
i 3
= a 2i
and then there exists another coordinate 1 for which
Pl (a )
3
= a 3l
~ a
2l
= Pl (a2 )·
In the first case the subset (l'(;k'Si'~j) gives the submatrix:
a
lk
ali
Sk
1
a
slk
k
a
a
a
lk
lk
a
2i
*
1
1
ali
4k
ali
*
1
lk
~k
.. a lk
~k
a
a
~i
1
~k
4k
ali
~i
~ a2i
i
a
~ li
i
ali
Si
~j
a
lj
~j
*
a
~ 3j
j
*
20
which is non-singular.
Its determinant is equal to:
+(~alk_ ~a4k)(S~2i_ Sali)(S~lj_
-
k
~
k
i
J
f 0
Sa3j )
j
In the second case the subset (l'~k'~i'Sl) gives the submatrix:
*
1
sk
~.
~
1
alk
Sk
ali
Si
*
alk
~k
~i
1
alk
Sk
~i
1
Again its determinant is equal
d)
a
1
a
*
~
4k
~k
to:(~:4k _
2i
~
1
a
21
1
a
S 31
1
~i
a .
S l~
*
i
s:lk)(s:2i_
~:li)(s~2l_~~3l)f O.
We are left with the case:
in k,
and
Pk(a l )
= Pk (a2 ) = alk
P (a )
k 3
= Pk (a4 ) = a 3k
But there exists another coordinate j such that Pj(a )
l
different from a
2j
= a lj
is
= P j (a 2 ).
) is different from a lj
j
4
3
and a
we are in the case a) or the case b). Also if P (a ) and
2j
j 3
P (a ) are both equal to one of the coordinates a
or a
we are in the
j 4
ij
2j
case c). Thus we only have to see the case where
If one of the coordinates P (a ) or P (a
j
(a and a have a symmetric role).
4
3
Then the subset (l'Sk,Sj,SjSk) gives:
21
a
a
a
1k
a
1k
1j
1
I;k
1
~
a
~.lj
J
1k
k
a
1k
a
~k
1
2j
a
a
3k
3k
Sk
1
1j
a
a
a
3k
1
2j
a k a ,
~ 1 ~ 2J
k
j
~~lj
a
a
3k 1j
~k ~j
J
~
3k
I;k
a
a
1k 1j
~k ~j
2j
~j
8
a
~j~k
CJ
a
j
2j
8
a
3k 2j
I;k I;j
Its determinant is equal to:
a.
a
2
+(S 2J_ ~ 1 j )
-
j
j
We shall now continue, but only with Abelian group with two invariants:
G
= Gh
~
1
G
h2
The characters will be denoted by I;
u
u v
~
= 0,1, .•• ,h1 -
1
v = 0,1, ..• ,h - 1 .
2
Then
Proposition 4.5
r I' 2 ,~,'~
I- 3 1,4
2 3 4, ) h as the P5-propert y.
.1Iue se t (1 ,~,~
,~"I) ,~ ,~ ,~~
I'm..
Consider 5 elements of G:
We note that there always exists a coordinate in which at least 3 e1ehave distinct coordinates.
Let us make the reasoning on the first co-
ordinate.
We have 4 cases to consider:
1.
2.
3.
4.
all a 21 a
31
all a 21 a
31
all a 21 a
31
all a21 a
31
a 41 a
a 41
all distinct
51
distinct and a
distinct and a
distinct and a
51
41
= a 41=
= a 31
* a
51
41
a
31
and a
51
= a 21
22
, h2 ,S,~
.3 ,.4) gives a non-singu1ar mat rix .
In case 1, the subset ( 1,~,~
f·2 ,s,~
h3 ) will give t h e submatrix:
In esse 2, the subset ( 1,~,~
1
*
1
*
1
*
1
~a31 ~2a31
1
~a41 ~2a41 ~3a41 ~a42
1
sa41
a
!;3 31
*
~2a41 !;3 a 41
and its determinant is equal to:
a
!; 11
1
1
a
which is different from 0 since
2
1
S 31
1
a
~ 41
2a
S
41
a52~ a 42 and al1a21a31a41 all different.
2
In case 3, the subset (l,~,s ,~,~ ) gives the submatrix:
~2
1
2a
*
*
S
1
11
28
21
*
*
*
*
1
!;
1
~ 2a31 ~ a 32
a
1
~ 31
1
~ 31
a
!;
2a
a
31 ~ 42
,2a
31
~
a
~
52
23
and its determinant is:
1
~
1
~
all
a
a
1
~
21
3l
2a
ll
2a
2l
~
2a
3l
a
~
•
1
'1
1
'1
a
a
~
1
'1
2a
32
2a
1') 42
2a
52
'1
32
'1
42
52
~ 0
since all a
a
are all different and also a
a
and a •
2l 3l
52
32 42
2
In the latter case we pick the subset (l,~,~ ,1'),~'1) and we
have
all
a
2l
*
a
22
1
S
1
S
~2
all
a
2a
~
2a
1
~
2l
~
1
S 3l
!;
a
a
a
a
3l
3l
2l
a
a
32
42
a
3l
~
52
1
S 2l
S
1
~
2a
2l
Its determinant is equal to:
~
a
a
a
a
a
a
(s 31 _ ~ 21)(1') 42_ 1') 32 )(1') 52 _ 1') 22)
1
1
2l
2a
3l
~
a
a
ll
2a
3l
1
~'1
1')
all
a
!; 2l
a
3l
S
*
*
1') 22
a
32
T}
~a3~a32
a
a
1') 42
a
52
'1
a
a
2l 22
~
'1
a3l a42
1')
a
a
!; 2l1') 52
~
2a
S ll
~
2a
21
~2a3l
~ 0
since all a 2l a
are all different and also a42~ a
and a52~ a 22 •
3l
32
Proposition 4.6
23452345.22
The set (l,~,s ,s ,~ ,~ ,1'),T) ,1') ,1') ,1') ,~1'),~1') ,~ 1')
has t~~_':6-
property.
Let a a2a3a4a5a6 be 6 d1atinct elements of G. We remark that there
l
always exists a coordinate in which at least 3 elements have distinct coordinates.
Let us make the reasoning on the first coordinate.
24
Thus we only have 6 cases
1.
all a 21 a
31
all a 21 a
31
all a 21 a
31
all a 21 a
31
all a 21 a 41
2.
a 41 a
a
all distinct
61
distinct and a61= a
51
a 41 a
51
51
a 41 distinct and a 61= a = a 41
51
a
distinct and a 61= a
and a 41= a
51
51
31
distinct and a = a 21
31
all a
a
distinct and a 21= all a 41 = a
a 61 = a
51
31 51
31
3·
4.
5·
6.
In case 1, the set
(1,~,~2,~3,~4,~5) gives
a non-singular matrix (of
Vandermonde) since all the coordinates are different.
4
In case 2 the set (1,~,~2,~3,~
,q) gives:
a
2a
3a 11
11
S 11
1
S
~
all
a
2a
3a 21
21
21
a
1
~
~
t
21 *
a
2a
3a 31
31
~ 31
a
1
~
~
31 *
a
3a 41
2a41
41
a
1
S
~
~
41 *
a
2a
a
1;3 51
51
a
a
1
t; 51
~
51 52
a
2a
3a51
51
51
t;
a
1
~
~
51 a62
'*
its determinant is equal to:
a
a
all
(q 62_ q 52) :
1
~
a
21
1
~
a
t; 31
1
a
1
S 41
a
t; 51
1
2a
~
11
2a
S 21
2a
S 31
2a
41
~
2a
51
~
~
~
t;
~
~
3a 11
3a 21
4a
~
4a
~
~
11
21
4a
31
4a
t; 41
*
*
*
a
4a
t; 51
q 52
4a
51
q 62
~
4a
~
4a
~
11
21
4,a
3a 31
S 31
3a 41
4a
3a51
*
~
~
41
4a
51
a
25
different from zero by our hypothesis.
In case 3, we take the
2 3 2 ) and we get the
set:(l,~,s ,~ ,~,~
submatrix:
*
1
*
*
*
1
*
*
*
1
*
*
1
1
1
and its determinant is equal to:
1
1
1
I: 0
1
1
1
1
since all a
a
a
are all different and also necessarily a
a
a ·
21 31 41
42 52 62
2 3
In case 4, we take the set (l,s,~
,~,~~) and we obtain the
,s
submatrix:
all
a
a
a
a
a
21
*
*
1
1
1
31
1
31
1
51
51
\
1
'*
*
26
Its determinant is equal to:
1
a
!; 11
a
2l
1
S 31
1
a
I; 5l
1
a
a:
a
a
a .
a
~(T} 42_ 1) ~2)'(Tl 62_ 1}5 2 )(1; ~l_ !; 31 )
~
2a
~
ll
~2a2l
a
2a
3l
2a
l; 5l
l;
3a ll
S
a
s3 2l
~3a3l
I;
3a
5l
which is different from zero since all a2l a
a
are all different
3l 5l
and a42~ a 32 and a62~ a 52 .
2
2
In case 5, the set (l,s,1; ,1),1) ,1;1)
is chosen and the following
matrix is obtained:
all
a
a
a
a
a
2l
2l
4l
4l
4l
1
*
a
a
a
a
a
l;
all
a
22
32
42
52
62
1
S 21
1
~
a
1
1
1
Its determinant is equal to:
2l
a
1; 4l
a
S 41
a
4l
~
2a
S
ll
2a
1; 2l
2a
!; 2l
~2a4l
2a
S 4l
2a
1;
4l
*a
*2a
22
Tl
2a
T} 32
2a
T} 42
2a
T} 52
2a
62
T}
1) 22
a
T} 32
a
T} 42
a
T} 52
a
T} 62
1 l;
all
1;
*a
a'
1; 21T},22'
a
a
l; 2l T} 32
a
a
!; 4l T} 42
a
a
!; 4l1) 52
a
a
4l 62
2a ll
a
a
a
a
a
a
a
a
a
a
a
2a
~(T} 32 _ 1) 22)(T} 5 2 _ 1) 42)(T} 62_ T} 42)(T} 62_ T} 5 2 )(1; 41_ S 21) 1 !; 2l I; 2l
a
2a
1 !; 4l 1; 4l
all a 2l a 4l are all different
a 42 a 52 a 62 - - - - - - -
a32~ a 22
1]
1;
I:
0
27
,,~.2 ,~,~,~
S S2) and we h ave:
In case 6 we take the set ( 1,~,~
all
a
all
a
a
a
31
u
u
a
a
31
a
51
u
51
~
1
12
11,
a
,. 11
1
22
a
<:,
32
S 31
a
1; 31
a
~ 51
a
S 51
1
42
1
52
1
62
11
,,2a11
~
I
:>
a
1
~2a11
~2a31
2a
a
22
11
a
51
Its determinant is equal to:
a
a
a
a
a
~
a
all
1
i;
1
S 31
a
~(~ 22_ ~ 12)(11 42_ 11 32){~ 62_ ~ 52)
a 22 all
a
11 32 1; 31
a
u
11 42~ 31
a
a
11 52 i; 51
a
a
62 51
~
1;
11 32
u
42
~
u
52
~
a
62
11
2a
i; 31
i;2a51
1;
0.
a
11121; 11
0,12
!;
a
1;
a
1
2a
~a12s2al1
a
11
22
11a32~~a31
a
~
42
2u
S 31
a
2u
11 52!; 51
a
2a
~ 62i; 51
~
11
2a
31
2a
1; 51
1; 51
,2a
ll
::,
~
°
Let us conclude by an application and an example.
Proposition 4.7
eel
e
If g = (2 - 1)(2 _ 1), we can construct a matrix of g rows and
~(e1+
1) columns with entries from GF(2) whichhas the P -property.
2
ee
e
Consider the Abelian group G = (h ,h ),h = 2 1_ 1, h = 2 _ 1. Then
1
2
l 2
ee
if x is a primitive root of GF(2 1), the character of E (G,2,ee ) are:
l
~u v
ua +e va
~ ~
: (a ,a ) ---> x 1 l 2
l
2
= O,l,
,h -l
l
v = O,l,
,h -1
2
To n = 1 corresponds the equivalence relation
u
a~ a2 is an automorphism of GF(2), we have:
~(l)
and as
~:: i;2 mod ~(l)
11 :: 11
2
mod ~(1)
28
Henc~
GF(2).
by (1.2) and (4.2) the set
(~,~)
has the P2-property over
eel
e
On the other hand S belongs to GF(2
) and ~ to GF(2). Hence
by' (2.3) we can construct a matrix, with entries from GF(2), of grows
n~
= eel + e = e(e + 1) columns, which has the P - p rop erty.
2
l
ele+e-l
ele
e
ele+e
Moreover since 2
- 1 < (2
- 1)(2 - 1) < 2
- 1 for
and nt +
e > 2, the matrix obtained from the Bose-Chaudhuri construction ~17
e(e +l)
l
) as an
page 73, by representing each non-null element of GF(2
e(el+l)
ele
e
e(el+l)-vector over GF(2) and deleting (2
-1) - (2
_1)(2 _l) rows,
has as many columns, that is e(el+l).
Let us work out the group G = (3,3)
p = 2 has the order m = 2 in the residue system modulo 3:
2
2
=4 •
1(3)
2
Hence if x is a primitive root of GF(2 ), the characters of
(a ,a )
l 2
u=O,1,2
~
are:
ua +va
2
x l
a =O,1,2
1
a =O,1,2
2
v=O,1,2
Now GF(22) has only GF(2) as a subfield (n=l).
the equivalence relation
~(G,2,2)
~(l),
the classes of characters are:
X*
o
1
X~
X;
~,
~2
~.
~2
X*
r
f2 2
X:
~~2,s2~
3
Corresponding to
~~,
~ ~
By (1.5)thegefive classes have the P - pro perty over GF(2) i.e. if
9
we pick one character from each class (1,~,~,~~,~~2) the obtained matrix:
29
1
~
1')
~1')
~T}2
(0,0)
1
1
1
1
1
(1,0)
1
x
1
x
(2,0)
1
x2
1
x
2
x
(0,1)
1
1
x
(1,1)
1
x
(2,1)
1
x
2
x
x
2
x
(0,2)
1
1
x
2
x
1
2
x
x
(1,2)
1
x
1
2
x
(2,2)
1
x
x
1
2
2
x
2
x
2
x
2
x
1
x
has the P9-property over GF(2).
2
Now the matrix representation of GF(2 ) of section 2 is:
x ~ (01)
11
x
2
~= 1 ~ (10)
01
~ (i~)
)
if we consider GF(22) as the algebraic extension of GF(2) by the
2
x + x + 1 .
addition of a root of the polynomial :
Hence if, in.the above'matrix, we make the substitution:
P:
xi(a) ~ P(Xi(a), n*
i )
(section 2)
the first column will remain alike and the elements of the other columns
will be replaced by a 2-vector over GF(2), namely the first rows of the
2
matrices corresponding to 1, x and x
1 ~ (1,0)
x
~
or
(0,1)
2
x
~
(1,1)
Then we get the non-singular matrix of order 9 with entries from
GF(2) :
30
A
=
(4.2) says that the set
1
~
1')
~1')
~1')2
1
1 0
1 0
1 0
1 0
1
o1
1 0
o1
o1
1
1 1
1 0
1 1
1 1
1
1 0
o1
o1
1 1
1
o1 o1
1 1
1 0
1
'1 1
o1
1 0
o1
1
1 0
1 1
1 1
o1
1
o1
1 1
1 0
1 1
1
1 1
1 1
o1
1 0
(~,s
2
2
,1'),1') ) has the P2-property.
Hence the set
* X* ) or simply (~,. 1') has the P -property over GF(2). Then
of classes (Xl'
2
2
the submatrix of A, formed by the four columns corresponding to Sand 1'),
has the P - property:
2
1 0 1 0
o 110
1 1 1 0
100 1
010 1
110 1
1 0 1 1
011 1
111 1
2
(l,S,~
'
2'
has the P - p roperty• Hence
5
* * *
property over GF(2). It
(X*
o ,Xl ,X2 ,X3 ) or simply (1,~,1'),~1') has the P5 implies that the submatrix of A, formed by the first seven columns, has
Finally (4.5) implies that
,1'),1') ,S1')
31
the P - property:
5
1101010
1 0 1 100 1
1111011
1100101
A
5
=
1 0 101 1 1
1 1 101 1 0
1101111
1 0 1 1 1 1 0
1 1 1 1 101
/
REFERENCES
1:17
LY
Bose, R. C. and Ray-Chaudhuri, D. K.,"On a class of error correcting
binary group codes", Information and Control, vol. 3, No .1,
March 1960, pages 68-79
--Bose, R. C. and Ray-Chaudburi, D. K.,"Further results on error
correcting binary group codes," Information and Control,
vol. 3, No.3, September 1960, pages 279-290--.--
LL7
Bose, R. C., "On some connections between the design of experiments
and information theory," To appear in Bulletin de l'Institut
International ~ Statistique.
l"1:.7
Peterson, W. W., "Encoding and error correction procedures for
Bose-Chaudhuri Codes," !:.!i:!. Trans. ££ Inform. Theory,
September 1960.
l"27
Van de.r Waerden, B. L., Modern Algebra,
Pub. Co., New York.
vol. 2, Frederick Ungar
Zierler, Neal,IIA class of cyclic, linear error-correcting codes in
pm symbols,1I M.LT. Lincoln Laboratory Group report 55-19.
ACKNOWLEDGEMENT
I wish to express my deep gratitude to Professor R. C. Bose who
taught me Coding Theory and its applications to Design of Experiments for
suggesting the problem to me and for his guidance and encouragement throughout the preparation of this work.
INSTITUTE OF STATISTICS
NORTH CAROLINA STATE COLLEGE
(Mimeo Series available for distribution)
258. Hoeffding, Wassily.
On sequences of sums of independent random vectors.
259. Webster, J. T., A. H. E. Grandage, R. J. Hader, R. L. Anderson.
dent variate in a linear estimator. June, 1960.
260. Chakravarti, I. M.
A decision procedure for the inclusion of an indepen-
On some methods of construction of partially balanced arrays. July, 1960.
261. Roy, S. N. and R. Gnanadesikan.
On certain alternative hypotheses on dispersion matrices.
August, 1960.
262. Murthy, V. K. On the distribution of averages over the various lags of certain statistics related to the serial correlation
coefficients. August, 1960.
263. Anderson, R. L.
Some needed developments in multivariate analysis.
264. Chapman, D. G., W. S. Overton and A. L. Finkner.
August, 1960.
Methods of estimating dove kill.
October, 1959.
265. Eicker, FriedheIm.
Consistency of parameter-estimates in a linear time-series model.
266. Eicker, FriedheIm.
1960.
A necessary and sufficient condition for consistency of the LS estimates in linear regression.
267. Smith, W. L.
On some general renewal theorems for nonidentically distributed variables.
268. Duncan, D. B.
1960.
269. Bose, R. C.
October, 1960.
October, 1960.
Bayes rules for a common multiple comparisons problem and related Student-t problems.
Theorems in the additive theory of numbers.
270. Cooper, Dale and D. D. Mason.
271. Eicker, FriedheIm.
October,
November,
November, 1960.
Available soil moisture as a stochastic process.
December, 1960.
Central limit theorem and consistency in linear regression.
December, 1960.
272. Rigney, Jackson A. The cooperative organization in wildlife statistics. Presented at the 14th Annual Meeting, Southeastern
Association of Game and Fish Commissioners, Biloxi, Mississippi, October 23-26, 1960. Published in Mimeo Series, January, 1961.
273. Schutzenberger, M. T.
274. Roy, S. N. and
January, 1961.
J.
On the definition of a certain class of automata.
N. Shrizastaza.
275. Ray-Chaudhuri, D. K.
January, 1961.
Inference on treatment effects and design of experiments in relation to such inferences.
An algorithm for a minimum cover of an abstract complex.
February, 1961.
276. Lehman, E. H., Jr. and R. L. Anderson. Estimation of the scale parameter in the Wei bull distribution using samples censored by time and by number of failures. March, 1961.
277. Hotelling, Harold.
The behavior of some standard statistical tests under non-standard conditions.
278. Foata, Dominique.
1961.
On the construction of Bose-Chaudhuri matrices with help of Abelian group characters.
279. Eicker, FriedheIm.
Central limit theorem for sums over sets of random variables.
280. Bland, R. P.
281. Williams,
J.
282. Roy, S. N. and R. Gnanadesikan.
April, 1961.
283. Schutzenberger, M. T.
285. Patel. M. S.
287. Konsler, T. R.
April, 1961.
A coding problem arising in the transmission of numerical data.
April, 1961.
May, 19tH.
Two problems in the theory of stochastic bl"anching processes.
May, 1961.
A quantitative analysis of the growth and regrowth of a forage crop.
288. Zaki, R. M. and R. L. Anderson.
ning over time. May, 1961.
May, 1961.
Equality of two dispersion matrices against alternatives of intermediate specificity.
Investigations on factorial designs.
286. Bishir, J. W.
March, 1961.
An evaluation of the worth of some selected indices.
On the recurrence of patterns.
284. Bose, R. C. and I. M. Chakravarti.
February,
February, 1961.
A minimum average risk solution for the problem of choosing the largest mean.
S., S. N. Roy and C. C. Cockerham.
February, 1961.
May, 1961.
Applications of linear programming tcchnillues to some problems of production plan-