Part of this work was done whi Ze this author was with the Department
of Statistics" University of North CaroZina at Chapel BiU. The research
done by him was partially supported by the Army Researeh Office" Durham,
under Grant No. DA-ARO-D-31-124-G910 and the United States Air Force Office
of SeJientific Research under Contract No. AFOSR-68-1406.
0 7 8 9 1 3
6 1 7 8 9 2
5 0 2 7 8 9
'+ 6 1 3 7 8
0
7
8
9
1
3
5
2
'+
6
6 5 '+
1 0 6
7 2 1
8 7 3
9 8 7
2 9 8
'+ 3 9
3 1+ 5
5 6 0
0 1 2
9
5
0
2
'+
7
8
6
1
3
8 7 1 2
9 8 2 3
6 9 3 '+
1 0 '+ 5
3 2 5 6
5 '+ 6 0
7 6 0 1
0 1 7 8
2 3 8 9
'+ 5 9 7
3
'+
5
6
0
1
2
9
7
5 2
'+ 3
3 '+
9 5
8 6
7 0
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'+ 6
5 0
6 1
0 2
1 3
2 '+
3 5
8 9
7 8
9 7
COM BIN A TOR I A L
MATHEMATICS
YEA R
8
February 1969 - June 1970
IRREDUCIBLE BINARY CYCLIC CODES OF EVEN DIMENSION
by
P. 11e 1sa rte
MBLE Research Laboratorry
Brussels 5 Belgium
J . -t4 • Goe tha 1s *
MBLE Research Laboratorry
Brussels" Belgium
Department of Statistics
University of North CaroZina at Chapel HiU
Institute of Statistics Mimeo Series No. 600.27
Hay
1970
IRREDUCIBLE BINARY CYCLIC CODES OF EVEN DIMENSION
by
P. Delsarte
and
J.-M. Goethals
MBLE Research Laboratozry
Brussels~ Belgium
*
MELE Research LaboratoI'!f
Brusse ls ~ Be 19ium~ and
Depazrtment of Statistics
University of North Carolina
I. INmJOOCTION •
A cyclic code
(xn-l)!g(x)
base field
GF(qk).
(n, k)
is called irreducible when the reciprocal
of its generating polynomial
GF(q).
g(x)
is irreducible over the
Such a code is shown to be isomorphic to a finite field
This isomorphism was used in [2] for the purpose of simplifying
the analysis of weight-distributions.
It is used here to characterize the
weights of irreducible binary codes of even dimension in terms of weights
of vectors of smaller length.
Finally the existence is shown of a class
of irreducible binary codes in which only two weights occur, and which contains the class discovered by McEliece [6].
2.
THE BASIC ISOMJRPHISM.
Cyclic codes of length
n
over a finite field
as ideals in the algebra K[x]!(xn-l)
*
K are usually defined
n
of polynomials modulo (x -1)
over K.
Part of this work was done while this author was with the Det>artment of
Statistics, University of North Carolina at Chapel Hill. The research done
by him was partially supported by the Army Research Office, Durham, under
Grant No. DA-ARO-D-3l-l24-G9l0 and the United States Air Force Office of
Scientific Research under Contract No. AFOSR-68-l406.
2
These ideals are completely specified by their generating polynomials
or as well by their associated polynomials
k
determines the dimension of the ideal.
here are those for which the block length
characteristic of the field
simple.
The polynomials
there exist polynomials
g (x)
and
h (x)
K,
and
respectively,
e(x)'" m(x)g(x)
to the code generated by
erties of
n
is relatively prime to the
K[xl/ (xn-l)
semi-
hex)
are then relatively prime and
m(x),
which are relatively prime to
such that
=
m(x)g(x) + l(x)h(x)
The polynomial
and
whose degree
The only cyclic codes considered
which makes the algebra
g(x)
lex)
h(x)'" (xn_l) Ig(x),
g(x),
1
mod(xn-l).
(1)
is a multiple of
g(x)
and thus belongs
From (1), one deduces the following prop-
g(x).
e(x):
e(x)
mod h(xL
1
(2)
(3)
As shown by MacWilliams [4], each cyclic code contains an unique element
e(x)
having properties (2) and (3), which can be used as generator for the
It is called the idempotent of the code.
code.
Its importance is emphasized
in the following theorem.
THEOREM
K[x]!h(x)
1.
to
The mapping a(x)
the code
-+
a(x)e(x)
g(x)K[x]/(xn-l),
is an isomorphism from
that is the ideal generated by
g(x) ... (xn-l)/h(x).
PROOf:
Let
K[x] Ih(x)
a(x)
and
b(x)
of polynomials
be any two distinct elements in the algebra
modh(x)
a(x)e(x)
-
over
b(x)e(x)
K.
Then, from
mod(xn-l)
(2),
3
would imply a(x):: b(x) mod hex),
into.
a contradiction.
Hence, the mapping is
Since, from (3), one has
[a(x)e(x)] [b(x)e(x)]
•
mod(xn-l) ,
a(x)b(x)e(x)
and since obviously
a(x)e(x) + b(x)e(x)
the mapping is an homomorphism.
g(X)K[x]/(xn-l)
=
[a(x) + b(x)]e(x),
Finally, since
have the same dimension k
K[x]/h(x)
and
(the degree of hex»~ over K,
the mapping is one-to-one, hence an isomorphism.
Q.E.D.
It is well known that, when hex)
the algebra K[x]/h(x)
extension of K.
CoROLLARY
of degree
2.
is irreducible over K
is a faithful representation of GF(qk),
GF(q),
the k-th
Hence, the following corollary is proved.
A <JYcZic code defined by an irreducibZe po7,ynomia7, hex)
k is isoTOOrrphic to a finite field
GF(qk).
Such a code is called an ipreducibZe cyclic code.
k
GF(q)
III
Its isomorphism with
can be exhibited in another way, using the so-called Mattson-Solomon
mapping [5].
Let
and an ... 1.
If K'" GF(q),
a be any root of hex);
then a
then K[a] ... GF(qk),
extension field of K containing a.
belongs to
k
GF(q)
that is the smallest
To each element
c belonging to K[a],
we associate the polynomial
+ v n- I x
where vi is defined to be
n-l
,
(4)
4
(5)
and where
k-l qj
Tk(z)
EjeO
Q
Z
k
GF(q)
is the trace from
to
GF(q).
Then,
as shown in [3], the mapping
~
K[a]
C €
v (x)
c
K[x]/(xn-l)
E
(6)
k
GF(q).
is an algebra isomorphism between two models of
K[a]
where
a
is a root of an irreducible polynomial
One model is
hex)
e(x),
K.
One easily verifies that (6) maps the element
the idempotent of the code, and that, for
=
v (x)
c
a(x)e(x)
C"
k,
n
(x -l)/h(x)
and the other is the irreducible cyclic code generated by
over
of degree
I
a(a),
in
K[a]
onto
one has
mod(xn-l),
(7)
from which it follows that
c
..
Ra-1ARK:
a(a)
..
v (a).
(8)
c
The mapping (6) can be applied to any extension field of
which contains
a,
K
in which case it is no longer an isomorphism, but rather
an homomorphism.
3.
WEIGHTS OF IRREDUCIBLE BINARY CYCLIC CODES.
Although many results apply to the q-ary case, only binary codes will
be considered.
divisor of
n
Let
k
2 -1.
w be a primitive root in
Then
s
w .. a
k
GF(2 )
and let
s
be any
is a primitive n-th root of unity, where
is defined to be
(9)
5
GF(2 k)
Us i ng (6) ,
GF(2)
can be mapped onto the cyclic code of length n
over
whose nonzero vectors are the n-tuples
(10)
whose components (5) are defined via the trace Tk(Z)
GF(2).
i
Let us denote by w(w)
since each component is
k
from GF(2)
to
the weight of the vector (10), that is,
or l, the summation over the integers
0
n-l
i
w(w )
i Sj
Tk(w - ) .
~
•
(11)
jaO
Obvious ly, one has
(12)
It suffices thus to consider the values of
i
from 0
to
s-l
in order
to know the complete weight-distribution of the code.
From now on, we consider only even values of k,
we define
6 and y
say
k· 2m,
and
to be
(13)
It is easily verified that
and
y
By = w,
is a primitive' (2 m+l)-th
Suppose
s
divides
m
where
B
is a primitive root in
GF(Zm)
root of unity.
2 -1 and let
n
2
be
(14)
The field
GF(2 m)
can be mapped onto the set of n 2-tuples
... ,
(15)
6
whose weights we denote by
w(sj).
The rest of this section will be
devoted to the proof of the following theorem.
3.
THEOREM
2m_I"
If s divides
code of Length
then the UJeights of the irreducibLe
are given by
(22m_ I ) /s
s-l
=
w(wi )
I
2
w(Bj)w(Si-j).
(16)
j=O
The proof requires several preliminary steps, 1;vhich we state as
Lemmas 4 and 5.
~4:
2m-I
n
1 -i )]
[z-(y+y
...
z
2m- l
m-l
+
i=l
PRooF:
r
z
2m- 1 _2 j
j=O
~ (z)
Let us denote by
the above right-hand polynomial.
We first
show that
F (x)
...
m
x
1s congruent to
m,
-1
~(x+x)
1 + x + x
2
m
+ ••• + x 2 mod 2. We prove it
observing that the result is true for
2 2m- l _l
x(l+x )
congruent to
for
2m- 1
F (x) ...
m
+ F l(x) , where the first term of the right-hand member is
m2m_l
3
5
x + x + x +
+x
• So that, if the result is true
i
~ (y +y
argument [1] that
mod (2 m+l).
and that one has
2
m-l, it is also true for
implying
m'" 1,
by induction on
-i
)
=0
0,
m.
for
yi + y-1
=
A consequence is that
i '" O.
Now,
yj + y-j
•
m
hence
it can be shown by the BCH
is impossible unless
For otherwise the polynomial
v(x)
F (yi) ... 0,
m
m
xi + ~ + x 2 +l-i + x 2 +l-j
i
= ±j
7
y -2 , y -1 , 1, y, y 2
would have as roots the elements
at least
~l
2
a contradiction.
6,
Therefore,
i
distinct factors
[z - (y +y
~(z)
having the same degree as
~
Qm(z)
and thus have weight
being divisible by the
is divisible by their product, which
)]
must be indentical to it.
Q.E.D.
5
Th e set
ILCrrM.
--..
{yi+y -i :
set of elements of GF(2m)
PRooF:
i
a
1,2, ... ,2~1}
is identical to the
fJJhose inverse have trace unity in GF(2).
It suffices to observe that
~(z)
where
T (z)
is the trace from
m
m
GF(2 )
to
GF(2).
The result then fol-
lows from Lemma 4.
Q.E.D.
PRooF OF THEOREM
with the element
i
3.
The weight w(w)
in
GF(2 t1)
wi
2
of the codevector associated
is given by (11), which using (13) can
be expressed as
n 2-l 2
... I
w(wi )
vaQ
m
I
\T2m(8i-SVy-j)I,
jeaQ
is the trace from GF(22m)
where T (Z)
2m
m
for
8€GF(2),
from
GF(2t1)
one has
to
(17)
T (By)
2m
to
GF(2).
= Tm[8(y+y-l)],
Using the fact that
where T (z)
m
is the trace
(17) can be rewritten as
GF(2),
m
n -12
i
w(w )
2
I
I
v-Q
j"'O
\T [e1 - SV (yj+y-j)]\,
m
which, using Lemma 5, is equivalent to
(18)
8
(l9)
where the second summation runs through the
which have trace
2m-l
elements of
m
GF(2)
But, this last summation can be replaced by
1.
2m_2
2 ITm(8 i - s'1 B-.e., I . ITm<tl) I ,
laO
since T (Bl )
m
j
is either 0
runs from 0
(s-l)
to
or
and
1,
~
and replacing l
from 0
to
by
(n2-l),
(j-s~), where
finally gives
6-1
w(w i }
2
a
L L 2 ITm(S(i-j)-SP)I
jaO P
where
P
'1-~.
a
• ITm(Bj-s~)I,
~
But now, the summations in P and ~ are easily recognized
and w(B j ) respectively, hence proving the result.
to be w(Si-j)
Q.E.D.
REMARK:
Let
be the circulant matrix of order
W
m
s
whose first row
is given by
Wm
co
.
~c[w(l},
2
s-l
w(8}, w(8 ), ••• , w(8
}],
and similarly
W
2m
a
~C[w(wi); (1
co
0,1,2, ••• ,s-1)].
Then Theorem 3 is equivalent to
W
2m
III
2if,
m
(20)
9
from which, by induction, one obtains
(21)
fxAr.pLE:
We briefly discuss hereunder two simple examples which em-
phasize that the mapping of
not be an isomorphism.
root in
4
GF(2).
the triples
for
Ws
j.
.. 2~
m
GF(2)
onto the set of n -tup1es (15) need
2
For both examples
For the first, we choose
T{Sj), T(8 j - 5 ), T(Sj-10)
0,1,2,3,4;
hence W
4
a
5
4
example, we chose s · 2 _1 • 15,
as divisor of
have respective weights
so that
24
4
2 -1;
0,2,2,2,2,
and thus from (20),
which shows that the
code has only two distinct weights, aame1y
W4 •
s
B is a primitive
and
= ~(O,2,2,2,2),
• cUtC.(32,24,24,24,24),
reduces to the trace
m· 4
and
n • 1
2
(51,S)
32.
irreducible
For the second
and w(sj)
simply
Hence, we have
~c.(0.0.0,l,O,O.l,l.0,l,0.1.l,l,l),
and from (20) we get
W
a
•
ciJtC.(S,a,a,10,a,12,10,6,a,10,12,6,10,6,6),
which gives the weight-distribution of the irreducible
(17,S)
namely:
weight
number of vectors
0
6
S
10
12
1 4x17 5x17 4x17 2x17
code,
10
4. Two-wEIGHT IRREWCIBLE CfWC caES.
The
(5l.8)
irreducible code discussed hereabove was shown to contain
vectors of only two distinct weights.
It was already shown by McEliece
[6] that this property holds for all irreducible codes of length
s
Z is a primitive root.
is a prime for which
(9). when
This class of codes is
extended here. as a result of the following theorem.
6.
Let
s
multiple of r,
say
k'" Zmr.
THEOREM
n ... (Zk -1) /s
W
o
and w
be any divisor of zr+l ,
and dimension
k
and Zet k be an even
Then the irreducibZe code of length
over GF(2)
has onZy two distinct weights
which are the unique solution of the foZZowing equations
l
CD
Zk-l
(2Z)
Z + (s-l)w 2 ...
W
o
PRooF:
(i)
(Zmr+l )
1
{n+1)Zk-2 •
We first prove the result for
so t h at
n
can b e f actori ze d as
m odd. in which case
n'" n n •
l 2
Then. ([3]. Theorem 1) the code
isomorphic to the code
(n ,2)
l
over
GF(Zmr).
wh ere
(n.k)
n
over
s
divides
(2mr+l)/s
1 '"
GF(2)
is
For any vector
(23)
of this last code, the corresponding vector of the binary code
permutation-equivalent to the vector of length
-Vi
is obtained from
[T
6 €GF(2mr )
i
is
n n
l 2
.... Vn 1 - 1)'
where
(n.k)
(24)
by the mapping
nZ-l
(6 ). T (6 a) ••••• T (aia
)].
mr i
mr i
mr
(Z5)
11
where
is a primitive root in
a
which correspond to nonzeromr l
2 -
GF(2
~i
y
).
But,
have weight
times the weight of (23).
is the image, and if
mr
If
since all vectors
mr-l
2
,
(25)
the weight of (24) is
w is the field element of which (23)
is a primitive
nl-th root of tmity, then
6
i
is given by
£3
i
=
i
which is zero iff
n
n -1·
1 '
or
l
and it is
n
+
(wy - )
wy
-i
it is
n -1
1
(n,k)
of the binary
,
(26)
mr ).
Consequently, the weight of (23) is either
when
W€yi[GF(2 mr)]
that is
(s-1)n n
1 2
€GF(2
otherwise,
l
mr
-i 2
(wy)
which occurs
n n
1 2
times,
Hence, the weights
times.
code are
(27)
w
..
1
which occur
case
2
n ,
1
times and
For
(11)
2k /2_1
n
mr-l
(a-l)n
times, respectively.
m even, we prove it by induction, observing that
in that case, so that Theorem 3 applies.
k = 4mr
s
divides
Hence let us consider the
and suppose the. theorem is proved for
k" 2mr, in which case
we have
(28)
where
w
o
I
and
and
w
J
are the tmit and all-one matrices of order
are given by (24).
1
s, and where
Then, from (20) we get by straightforward
calculations
W
2m
where
n" (2
..
2
2mr-1
2mr-1) Is,
[(n+l)I + n(J-I)],
(29)
hence proving that the only two weights occurring are
12
W
o
=
w1 •
22mr-l (n+l),
(28)
22mr- l n.
This completes the proof since (27) and (28) are the general solutions
of (22) for the cases
k
D
2mr,
and k. 4mr
respectively.
Q.E.D.
Some numerical results follow (Table 1).
13
TABLE 1:
TbJo-weight cycZic codes
r
k
s
1
1
3
4
3
1
3
6
8
1
3
3
10
12
5
5
8
12
12
16
20
1
2
2
3
4
9
17
33
n
W
o
I
WI
5
21
4
85
48
341
1365
160
704
40
176
672
51
819
32
384
24
416
455
3855
256
2048
224
1920
31x1025
32x512
93
63x4097
315
32
64x2048
31x512
48
63x2048
128
160
8
2
12
5
5
11
6
6
65
13
10
24
12
7
9
43
27
14
18
381
19x511
128
18x256
192
19x256
9
171
18
2x256
3x256
9
57
18
3x511
gx511
8x256
9
10
10
19
205
41
18
20
20
11
12
683
241
22
27x511
5xl023
25xl023
3x2047
26x256
4x512
24x512
2xl024
9x256
27x256
5x512
24
17x4095
16x2048
25x512
3xl024.
17x2048
14
PfFERENCES
[1]
BOSE, R.C. and RAY-CHAUDHURI, D.K., On a class of error correcting
binary group codes, Information Controz.~ 3 (1960), 68-79
and 279-290.
[2]
GOETHALS, J.-M., Algebraic structure and weight distribution of
binary cyclic codes, Uni ve1'8i ty of North Caro Una, Ins titute
of Statistics Mimeo Series No. 484.4 (1966).
[3]
GOETHALS, J .-M.,
Factorization of cyclic codes,
I.E.E.E., Trans.
on Information Theory, 11-13 (1967), 242-246.
[4]
MacWILLIAMS, F.J., The structure and properties of binary cyclic
alphabets, Bel.l.SystemTechn. J., 44 (1965),303-333.
[5]
MATTSON, H.F. and SOLOMON, G., A new treatment of Bose-Chaudhuri
codes, J. Soc. Indus. AppZ.. Math., 9 (1961), 654-669.
[6]
McELIECE, R.J.,
Jet P1'OpuZ.sion LaboraVol IV, 264-266.
A class of two-weight codes,
tory Space Programs Summary
37-41~