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ALGEBRAIC STRUCTURE A1ID WEIGHT DISTRIBUTION
OF BINARY CYCLIC CODES
by
J. -M. Goethals
M.B.L.E.
Laboratoire de Recherches
Bruxelles 7, Belgium
Institute of "Statistics Mimeo Series No. 484.4
August 1966
Presented at the NATO Summer School on
Combinatorial Methods in Coding and
Information Theory, Royan, France
August 26 - September 8, 1965
Organized by:
The Department of Statistics
University of North Carolina
and
The Institute of Statistics
University of Paris
This research was supported by the Air Force
Office of Scientific Research Contract No.
AF-AFOSR-76o-65.
DEPAR'IMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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Algebraic structure and weight distribution
of binary cyclic codes
Introduction
Some algebraic properties of cyclic codes are presented which are useful in the analysis of their weight distribution.
are found in tw recent papers by Me Williams [1] and Nili [2].
Results on computer analysis of weight distributions in binary cyclic
codes are also given.
up to 43.
The
ana~sis
is exhaustive for all odd block length
It is to be noted that most of these results have been
obtained by Kilman
recent~
Goldman and Smola [5], without any simplification in the
analysis, with the aid of a very powerM computer.
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Some ideas developed here
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1.
.1
Algebraic structure of cyclic codes
1.1.
I
A cyclic code over a finite field GF(q) may be considered as an
n
ideal in the ring of polynomial residue classes mdulo (x -1) over GF (q) •
The mnic polynomial g(x) of minimum degree in such an ideal is known to be
a divisor of (xn_l) and is called "generator" of the ideal.
If r denotes its
degree, then the dimension of the ideal is given by
k
which is also the degree of
=n - r
(1)
the reciprocal factor of g(x), i.e.
Let us denote by
A
A[h(x)]
a cyclic code (n,k) over GF(q), generated by g(x).
:
the ring of polynomial residue classes mdulo hex), given
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by (2).
Assuming n and q to be relatively prime, the polynomials g(x) and hex)
are then relatively prime, and there exist polynomials m(x) and n(x) such
that
m(x) • g(x) + n(x) • hex)
=1
with m(x) and n(x) relatively prime to hex) and g(x), respectively.
As a well known result in the theory of rings and
complete isomorphism between A and A[h(x) J.
idealS~~
there is a
This isomorphism is character-
ized as follows.
The element
e(x) .. m(x)·g(x)
(4)
being a multiple of g(x), is a vector of the c~de A and has the following
el
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properties:
~)
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See f.i. ref. 3 or 4
2
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e(x)
jil
1
e 2 (x) =' e(x)
mod. h(x)
(5)
mod. (xn -1)
(6)
i.e. e(x) is an unit in A[h(x)J and idempotent in A.
Now let
v(x)
= a(x)·e(x)
(7)
be a vector of the code A, with a(x) an element of A[h(x) J.
It is easy to
see that the correspondence
a(x) in A[h(x)J ~ v(x) in A
(8)
is actually an isomorphism.
1.2.
When the polynomial h(x) of degree k is irreducible, the ring
A[h(x) J is known to be a field, namely the Galois field GF(qk).
plicative group of this field is known to be cyclic, i.e. there exists an
element a(x) called "primitive" such that
1, a(x), r1(x),
... ,
are all the distinct non-zero elements of GF(qk).
As a consequence, all the elements of the code A may be represented as
~(x) ·e(x),
(i • 0, 1, 2, ••• qk -2)
(10)
i.e. may be generated by successive multiplications by a(x).
,
Let n be the exponent to which h(x) belongs, Le. the least integer
such that
>f
,
,
-1 be divisible by h(x).
Then, necessarily n
divides qk -1, i.e.
k'
q -1 • n ·m
(11)
_
m
and a-(x)
must
be of order'
n , i.e. must be a primitive root of (xn' -1).
Let us write
cf1(x) ~ x J mod h(x)
,
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The multi-
(12)
where j is relatively prime to n , and m is the least integer such that
(12) holds, since otherwise a(x) would not be a primitive root.
3
II
.11
As a consequence of (12)
II
and more generally
aF+i(x).e(x) = x j
at(x).e(x) mod (Xn_l)
(14)
which shows that at(x) .e(x) for i ~ m are simply shifted repetitions of the
first m ones.
It has been proved.oo that all vectors in such a code have the
,
same cycle length n , so that the weight distribution of the code is simply
given by n
,
times the weight distribution among the m vectors
e(x), a(x).e(x), oF(x).e(x), ••• , aF-l(x).e(x)
1.3.
(15)
In the general case where h(x) is the product of several irreduci-
ble factors
h(x) • h (x).h2 (X) ••• hs(X)
l
of respective degrees
~, ~,
••• k ' it has been
s
Shown~**} that
(16)
A is the
direct sum
A=~+~+
•••
(17)
of the ideals Ai generated by
(18)
The ideals Ai and their idempotents e (x) are called primitive, since they
i
cannot be decomposed into the direct sum of some subideals; they are known
to be isomorphic to a field.
Furthermore, they are mutually orthogonal, i.e.
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II
II
ell
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(*) See ref. 1, proposition IX.
(**) See ref. 1, lemma 2.12; we use the term "direct sum" rather than "union"
as it is called there.
4
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as it may be seen from (18).
It is easy to see that A contains every subsum
+• A
A.
1.
1
i
+•
2
...
with
i.e. (2 s _1) different subidea1s, A included.
The idempotent of A is given by
(21)
where ei(x) is the idempotent of Ai' since from
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(20)
(22)
it follows that (21) is congruent to 1 modulo hi (x) for i = 1,2, ••• , s, and
thus modulo h(x).
We now show that the set G of elements of A that are contained in none
of the subidea1s (20) form a multiplicative group whose order is given by
k
~
k
(q 1 -1) (q
-1) ... (q s -1)
First, these elements are all of the form
v
= v1
+v
2 + ••• + Vs
(24)
k.
where vi is a non zero element of Ai' and since there are (q 1._ 1 ) non zero
elements in Ai' their number is given by (23).
5
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Now, since Ai is a field, every non zero element Vi of Ai has an inverse·
-1
Vi
I
in Ai such that
.1
and thus
v
-1
-1
• vl
-1
+ v2
-1
+ ••• +vs
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(26)
is an element of G such that
l
••• + v s v-s
+ ••• + e s
(27)
-e
el
so that G is actually a group.
It is easy to show that all the elements of G have the same cycle
t
length n , where n
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t
is the least common multiple of the exponents to which
the hi (x) belong.
t
By hypothesis, n
is the least integer such that (Jf -1) is divisible
"
by (16), so that it is impossible to have for vex) in G, and n < n
t
"
"
since otherwise either vex) would be in a subideal, or (Jf -1) would be
divisible by (16).
I
It is now easy to see that x.e(x) generates a subgroup H
n
t
of order n
in G, and that the factor group G/H t contains all the cycle representatives
n
of G. This group turns out to be in general non cyclic so that a "system of
6
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generators" is needed in order to generate the whole set of cycle representatives in a multiplicative way, similar to that used in (10).
2.
Computer analysis of weight distributions in binary cyclic codes
The ideas developed in section 1 have been used in the analysis of
weight distributions in several binary cyclic codes.
(17), the program generates the cycle representatives of the group G of regular elements for each subideal (20).
serving operation over the binary field, so that the cycle representatives
may be divided into automorphism classes, each class containing elements that
are obtained from each other by squaring..
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codes of odd block length up to 43.
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Further reduction in the amount of
computation has been obtained using the fact that squaring is a weight pre-
need to be considered.
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For an ideal A such as
Only one element from each class
An exhaustive analysis has been done of all inequivalent binary cyclic
For codes of dimension k greater than
(n-l)/2, the weight distribution has been obtained directly from the null
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code using Mc Williams formula
(*)
•
These results together with some additional results for block length
greater than 43, are given in section 3.
For each block length n, the
n
irreducible factors of x -1, given in octal form, are labelled f , f , f •••
l
3 5
according to the following rule: one of the primitive polynomials is labelled
f 1 j tllen another polynomial is labelled f j if, ex being a root of f l' j is the
smallest power of ex such that
not mentioned.
aJ
is a root of f j.
The code (n,k) labelled f
(~ See ref. 6
7
i
f j ... f
The polynomial f 0 = 1+x is
m
is generated by
I
and has only even weigh:t vectors; the code (n,k+1) generated by g/1' is not
o
mentioned, since it has the same even weight vectors as the one generated by
g and, in addition, the odd weight vectors obtained by adding the all-one
vector.
3.
Numerical results
3.1. Polynomials
n
7
9
15
17
21
23
25
27
31
33
35
39
41
43
1'1
13
111
23
727
127
5343
4102041
1001001
45
3043
16475
17075
6647133
52225
1'3
15
007
37
471
015
00001ll
75
3771
13627
17777
5747175
47771
1'5
1'7
07
31
165
6165
0000037
007
1'9
1'11
1'13
1'15
57
00037
13617
64213
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013
0000007
67
2251
00013
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.1
73
0007
51
p0015
I000o7
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3.2.
Weights
Codes
(7,3)
(9,6)
(9,2)
(15,4)
(15,4 )
(15,2)
(15,6)
(15,6)
(15,8)
(15,8)
(15,10)
(17,8)
(21,6)
(21,3 )
(21,2)
(21,5)
(21,8)
(21.9)
(21,9)
(21,8)
(21,11)
(21,11)
(21,12)
(21,12)
(21,14)
(21,15 )
(23,11)
(21,6)
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Codes
Polynomials
f
l
f1
f
3
f
1
f
3
f
5
f ,f
l 5
f ,f
3 5
f1'f 3
f ,f
l 7
f l ,f3'f5
f1
f
1
f
3
f
7
f3'f 7
f ,f
1 7
f ,f
7 3
f ,f
1 9
f3'f 7,f
9
f ,f ,f
1 3 7
f ,f ,f
1 7 9
f ,f ,f
l 3 9
f ,f
1 5
f 1,f ,f ,f
3 7 9
f ,f ,f
1 5 3
f
1
f3'f 9
8
0 2
4
6
1 0
1 9
7
27
0
27
1 0
1 0
10
12
5
0
0
3
0
15
1 0
0
10
0
0
1 0
1 0
0
0
0
0
30
15
3
18
0
30
75
3
60
1 0
1 0
15
25
100
1 0
30
60
105
60
1 0
105
280
,435
168
1 0
1 0
0
68
68
0
0
85
21
1
0
0
0
0
0
1 0
1 0
0
0
0
0
0
0
0
21
1
0
0
21
21
1
0
0
0
1 0
1 0
21
0
14
16 18
0
5
5
0
35
34
42
0
0
0
0
7
0
0
0
0
3
0
0
3
45
0
0
126
7
42
0
0
210
0
280
0
21
0
0
147
0
343
0
0
0
0
21
21
0
168
210
98
280
3
360
0
0
105
1008
7
0
1 0
1 0
21
0
297
343
1071
147
21
189
903
1 0
1 0
63
1197 1295
1260 1281
399
546
1
84
1
0
210
1
0
0
1
0
0
735
924 2982
1638 6468
0 506
210
21
9
0
21
147 21
84
7
0
0
5796 4340 1956 273 28
10878 9310 3570 651 42
0 1288
0 253
0
0
0
0
35
7
20
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Weights
Codes
(25,4)
(25,20)
(27,2)
(2'7,6)
(27,8)
(27,18)
(31,5 )
(31,10 )
(31,10)
(31,10 )
(31,15)
(31,15)
(31,15)
(31,15)
(31,20)
(31,20)
(31,20)
(31,25)
(33,10)
(33,10)
(33,2)
(33,12)
(33,12)
(33,20)
(33,20)
(33,22)
Polynomials
1'5
1'1
1'9
1'3
1'3'f9
l'
1
1'1
1'1,1'3
1'1,1'5
0
2
1
0
0
0
1
1
1
1
1 27
1 0
1 0
1 0
1 0
1 0
1 0
1 0
11000
0
65250
0
0
0
10206
1'1,1'15
f 1 , 1'3' 1'5
1'1,1' ,f
5 7
f l ,f3'f15
1
f l ,f3'f
7
f l' f , f , f
1
3 5 15
1
f 1, 1'3' 1'5' f 7
1
1'1' f3' f 7, f 15
1'1,f3,f5,1'7,1'11 1
1
1'1
f
1
3
1
1'11
1
f1' f i l
1
1'3,1'11
fl'f3
f 1,f
5
1'l ,f3' 1'11
1
1
1
4
0
50 1025
0
0
0
0
0
0
0
324
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 1085
0
0
0
0
0
0
0
0
0
0
0
0
0
0
165
0
10
6
8
9
36
2268
0
0
0
0
0
0
31
0
806
806
10
0
0
0
0
465
465
310
310
7905
7905
12
10
207500
0
0
0
30618
0
0
0
326250
0
27
126
61236
0
310
0
31
0
0
1116
4340
1271
41602
4340
142600
41602
142600
310
155
8680
8680
837
7595
42997 138880
22568 247.845 1383096 4414865
0
0
0
165
0
55
0
330
0
0
0
0
0
0
165
396
0
55
0
363
220
3795
21615
87648
6930
990
27720
84546
1276
13200
90453
347457
.
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Weights
Codes
Polynomials
14
(25,4)
(25,20)
(27,2)
(27,6)
(27,8)
(27,18)
(.31,5)
(31,10)
(.31,10)
(.31,10)
(31,15)
(31,15)
(.31,15)
(.31,15)
(31,20)
(.31,20)
(.31,20)
(.31,25)
(3.3,10)
(.3.3,10)
(.3.3,2)
(3.3,12)
(.3.3,12)
(.3.3,20)
(3.3,20)
(.33,22)
f
5
f1
f
9
f.3
f ,f
3 9
f1
f1
f 1,f
3
f 1,f
5
f 1,f
15
f 1, f3' f 5
f 1,f ,f
5 7
f 1,f ,f15
3
f 1,f3'f7
f1,f3,f5,f15
f 1,f3'f5,f7
f1,f3,f7,f15
f 1,f3'f ,f7, f 11
5
f1
f.3
f ll
f 1,f11
f.3,f 11
f 1,f
3
f 1,f
5
f 1,f3'f11
0
275000
0
0
0
78732
0
0
0
.310
0
0
8370
8060
251100
251100
257610
8280720
165
0
0
495
495
199815
180180
797775
11
16
18
20
0
128125
0
0
0
59049
31
527
527
217
18259
18259
9393
9.39.3
301971
.301971
294159
9398115
165
0
0
1155
1.386
284031
270765
1140777
0
.31250
.3
27
84
1968.3
0
0
0
155
0
0
5580
5890
195.300
195.300
201910
6440560
.3.30
462
0
1155
1452
25.3902
270600
101.3298
5
.3125
0
0
0
0
0
186
186
155
5208
5208
2852
2852
85560
85560
81840
2648919
165
0
0
528
165
139920
162393
557898
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Weights
Codes
Polynomials
22
(25,4 )
(25,20)
(27,2)
(27,6)
(27,8)
(27,18)
(31,5)
(31,10 )
(31,10)
(31,10)
(31,15)
(31,15)
(31,15)
(31,15)
(31,20)
(:51,20)
(31,20)
(31,25 )
(33,10)
(33,10)
(33,2)
(33,12)
(33,12)
(33,20)
(33,20)
(33,22)
f
5
f
l
f
9
f
3
f3'f 9
f
l
f
l
f ,f
l 3
f ,f
l 5
f ,f
l 15
f l ,f3'f
5
f ,f ,f
l 5 7
f , f ,f
l 3 15
f l ,f3'f
7
fl,f3,f5,f15
f l , f3'f , f
5 7
fl' f 3 , f , f 15
7
f l' f , f , f 7, f 11
3 5
f
l
f
3
f
ll
f ,f
l ll
f ,f
3 ll
f ,f
l 3
f ,f
l 5
f 1,f3'f11
i
24
26
28
30
32
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0
0
0
0
0
0
0
0
9
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
155
0
0
0
155
0
0
0
0
0
0
31
0
18910
2635
186
0
18910
2635
186
0
20305
2325
217
0
628680
82615
5208
33
0
0
0
155
0
165
0
0
0
11
3
201
0
0
0
0
0
0
0
0
3
165
0
0
11
47058
1155
0
0
11
44286
9405
0
0
0
190842
36630
4521
165
775
620
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'12
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Weights
Codes
Polynomials
0
(35,3)
f
(35,4)
(35,6)
f
7
f ,f
5 25
f ,f
5 7
f ,f ,f
5 7 15
f
1
f ,f
1 5
f 1,fi5
f 1,f
7
f ,f ,f
1 5 15
f ,f ,f
1 5 7
f ,f ,f
1 7 15
f l' f 5,f7, f 15
f ,f
1 3
f ,f ,f
1 3 5
f
1
f
3
f 1,f1,
fy f
13
f ,f
1 7
f ,f
1 3
f ,f ,f
1 3 13
f
1
f
1
f ,f
1 3
(35,7)
(35,10)
(35,12)
(35,15)
(35,15)
(35,16)
(35,18)
(35,19)
(35,19)
(35,22)
(35,24)
(35,27)
(39,12)
(39,12)
(39,14 )
(39,lh)
(39,24)
(39,24 )
(39,26)
(41,20 )
(43,14)
(43,28)
5
h
6
8
10
1h
:J2
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
21
0
0
0
0
0
420
0
0
0
35
0
70
0
35
0
70
490
1
35
1 35
1 210
1 665
1
0
1
0
1
0
1
0
1 23h
1
0
1
0
1
1
1
0
0
70
56
0
70
490
70
0
0
1260
490
525
490
4200
1701
,693 0
10h30
48426
595
2100 25655 1751hc
13020 182525 11~26880
0
0
0
0
78
0
0
0
117
0
0
78
1716 15015
77220
234
3393 1~04011.
819
0
0
0
0
0
301
13
15327 153075
0
1312
0
0
9116 117691
0
10
0
10
10
0
3255
0
3430
0
91j.60
1925
14770
33075
19180
88700
15435
7h980
198485 577825
801220 2275560
6538805 181231~20
156
0
0
715
1401j.
156
715
314028
39
936936
233935 .933894
964171 3652623
7585
33210
0
344
931294 4792909
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Codes
Weights
Polynomials
16
( 55,5)
( 55,4)
( 35,6)
( 35,7)
( 35,10)
( 35,12)
( 35,15)
( 35,15)
( 35,16)
( 35,18)
( 35,19)
( 35,19)
( 35,22)
(35,24)
(35,27) ."
(59,12)
(39,12)
(39,14)
(39,14)
(39,24)
(39,24 )
(39,26)
(41,20)
(45,14)
(43,28)
f
5
f
7
f ,f15
5
f ,f
5 7
f 5, f 7, f 15
fl
f l ,f
5
fl'f 15
f l ,f
7
f 1,f ,f15
5
f 1, f , f
5 7
f ,f ,f
1 7 15
fl,f5,f7,f15
f l ,f
3
f l ,f ,f
3 5
f1
f
5
f 1,f
13
fy f
13
f 1,f
7
f l ,f
3
f 1,f ,f13
3
f1
f
1
fl'f 3
I
18
20
22
24
0
0
0
7
0
0
0
0
0
0
0
0
0
0
35
70
35
0
0
7
385
350
105
105
0
1505
0
2100
0
0
0
15365
12502
0
1540
" 12005
0
16807
0
0
12845
23520
8330
6230
1260
67445
59010
58415
6160
19670
92715
172900
74592
56550
9415
102900
188860
66297
49490
10430
964520
1146460
762125
100660
372645
4412520 5155570 1449700
3993535
412265
31688755 35460320 25403707 11510100 3258255
0
1053
2028
0
858
0
1716
0
1287
0
1053
2028
5070
5148
858
858
5148
2145
5577
1521
2111967
5517800 4222218 3454508 1727193
2269683
3862872 4138719 5167190 1509027
9275877 :15112385 16955367 12353913 6192069
195160
97539
255266
232060
146370
1204
2107
5311
3311
3999
16184770 37106979 58671522 64182703 488lt9892
.1
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Weights
Cades
Polynomials
26
(35,3)
(35,11-)
(35,6)
(35,7)
(35,10)
(35,12)
(35,15)
(35,15)
(35,16)
(35,18)
(35,19)
(35,19)
(35,22)
(36,24)
(35,27)
(39,12)
(39,12)
(39,14)
(39,14 )
(39, 2h)
(39,24 )
(39,26)
(hl,20)
(43,lh)
(11-3,28)
28
30
32
0
f
0
0
0
5
f
0
0
0
5
7
f ,f
0
0
0
7
5 15
0
0
f ,f
0
5
5 7
f ,f ,f "
0
0
7
5
5 7 15
f
0
0
0
0
1
0
0
f ,f
0
35
1 5
0
0
0
f ,f
0
1 15
0
0
420
f ,f
145
1 7
0
f ,f ,f
140
1225
7
1 5 I5
0
180
f ,:f ,f
0
2730
1 5 7
f ,f ,f
0
3430
985
35
1 7 15
16730
7
35
1755
f1,f5,f7,f15
0
0
f ,f
4200
71540
1 3
2100
0
556640
52535
f 1, f3'f 5
0
f
0
0
0
1
f
0
286
0
0
3
0
0
f ,f
0
549
1 13
0
286
f ,f
0
3
3 13
0
0
0
f ,f
398580
1 7
f ,f
502164
12922
936
101829
1 3
1966617 410241
52741
3510
f 1, f3' f 13
11-10
f
60021+
16605
3034
1
0
f
301
301
1505
1
f ,f
25735801 9224962 2240343 351869
1 3
34
36
0
0
0
0
0
0
0
13
0
13
0
13
13
117
0
0
0
0
2494
32809
~
15
II
3.3.
.1
Admtiona1 results
II
number of vectons
weights
16
20
24
28
32
2)
and
and
and
and
and
Code (55,21); polynomial
h
.
weights
Code (63,13) ;
51
51
504 x 51
480 x 51
111 x 51
= f 1,f0
number of vectors
16 and 39
20 and 35
24 and 31
28 and 27
32 and 23
36 and 19
40 and 15
3)
30 x
160 x
35
31
27
23
19
polynomial h
weights
70 x
1048 x
5296 x
8080 x
3947 x
600 x
24 x
55
55
55
55
55
55
55
= f 1,f5,fo
number of vectors
24
10 x
and 39
32 and 31
40 and 23
63
49 x 63
6 x 63
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4)
Code (63,13) ;
polynorria1 h = f 1,f ,f
3 o
weights
number of vectJrs
24 and ) /
28 and 35
32 and 31
36 and 27
40 and 23
3 x 63 + 1 x 21 = 210
24 x 63
= 1.512
= 1.071
17 x 63
18 x 63 + 2 x 21 = 1.176
2 x 63
=
126
~()
5)
Code (65,131; polynomial h = f 1,fo
number of vectors
weights
26 and 39
28 and 37
30 and 35
32 and 33
34 and 31
36 and 29
38 and 27
40 and 25
6)
Code (89,12);
6 x 65
7 x 65
12 x 65
12 x 65
6 x 65
9 x 65
8 x 65
3 x 65
polynomial h
=
weights
number of vectors
x 89
11 x 89
1 x 89
40 and 49
48 and 2~1
56 and 33
11
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f 1, f o
17
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7) Code (91,13); polynomial h = f l
weights
36
40
44
48
52
number of vectors
4
6
12
15
8
and
55
and 51
and
L~7
and 43
and 39
91
91
91
91
x 91
x
x
x
x
8) Code (151,16); polynomial h = f l
weights
number of vectors
60 and 91
64 and 87
68 and 83
72 and 79
76 and 75
80 and 71
84 and 67
88 and 63
3
5
30
65
39
30
40
5
x 151
x 151
x 151
x 151
x 151
x 151
x 151
x 151
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References
[1]
J. Mc Williams,
bets", B.S.T.J.
"The structure and properties of Binary Cyclic Alpha-
44, 2, pp. 303-333; Feb. 1965.
[2]
H. Nili, "Matrixschaltungen zur Codierung und Decodierung von
Gruppen-Code", A.E.U. 18, 9, pp. 555-565; Sept. 1964.
[3]
N. H. Mc Coy, Rings and Ideals,
M.A.A.; New York (1956).
[4]
B. L. van der Waerden,
New York (1949, 1950).
[5]
M. IQ.iman, H. D. Goldman, and H. Smola, "The Weight structure of some
Bose-Chaudhuri Codes", (privately communicated).
[6]
J. Me Wi11iams,"A theorem on the distribution of weights in a syste...
matic COde," B.S.T.J. 42, 1, pp. 79-94; Jan. 1963.
t,lodern Algebra (2 voL), Fred. Ungar Pub. Co;
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The Carus Math. Monographs nO 8,
19