TOOLS FOR COSET WEIGHT ENUMERATORS 13
Note that in this case the code
E
is an irreductible cyclic code viewed as
a cyclic code of length n, even when n and s are not relatively prime (by
t
repetitions of each symbol). When
t
=
1,
E
is equivalent to the simplex code
whose all codewords have a same weight
..
In this case the corollary above
means that the weight polynomial of
D;t
does not depend on
/3,
i.e. on
x.
In
fact
L(y)
=
n- .and every element in Eisa shift of
y.
COROLLARY
3. If all codewords of the code E have same weight, every coset
of C of depth r (the same depth than C) have the same weight enumemtor.
We suppose now that
E
has more than one weight. The code Cj_ consists of
L separate sets:
U
(oj(Yl)
+Pm(p-
1)-r+ 1),
Yl
E
E, l
E
[l,L].
jE[O,n-1]
Each set is a union of cosets of same weight polynomial, since the RM-codes
are invariant under the shift. Assume that for each
l,
this weight polynomial,
the weight of
Yl
and
¢s(Yl)
are known. Then, by applying Corollary 2, we can
compute the values
L(y1)
for every
l
and obtain the weight polynomial of
D;t.
The most simple case appears when the code E is such that only two weights
occur. For instance, using the tools we present here, we treat in [10] the cosets
of depth 2 of the extended 2-error-correcting BCH codes.
REFERENCES
1.
E.F.
AssMus & V. PLESS On the covering radius of extremal self-dual codes, IEEE Trans.
on Info. Theory, vol. IT-29, n. 3, May 1983.
2. L.A. BASSALYGO, G.V. ZAITSEV & V.A. ZINOVIEV, Uniformly packed codes, translated
from Problemy Peredachi lnformatsii, vol. 10, N. 1, pp. 9-14, January-March, 1974.
3. L.D. BAUMERT & R.J. McELIECE, Weights of irreductible cyclic codes, Information and
Control 20, pp. 158-175 (1972).
4. S.D. BERMAN, On the theory of group codes, KIBERNETICA, Vol. 1, n. 1, pp. 31-39,
1967.
5. R.A. BRUALDI & V.S. PLESS, Orphans of the first order Reed-Muller codes, IEEE Trans.
Inform. Theory, Vol. 36, N. 2, March 1990.
6. P. CAMION, B. COURTEAU & A. MONTPETIT, Weight distribution of 2-error correcting
binary BCH codes of length 15, 63 and 255, IEEE Trans. on Inform. Theory, vol. 38, No.
4, pp. 1353-1357, 1992.
7. P. CAMION, B. COURTEAU & P. DELSARTE, 01f r-partition designs in Hamming spaces,
Applicable Algebra in Eng. Comm. and Computing 2, 147-162 (1992).
8. C. CARLET Codes de Reed et Muller, codes de Kerdock et de Preparata, These de
l'Universite PARIS VI, Janvier 89.
9. P. CHARPIN, Codes ideaux de certaines algebres modulaires, These de Doctorat d'Etat,
Universite Paris 7, 1987.
10. P. CHARPIN, Weight distributions of cosets of 2-error-correcting binary BCH codes, ex-
tended or not, IEEE Trans. on Info. Theory., to appear.
11. P. CHARPIN, Distributions de poids des translates des codes BCH binaires 2-correcteurs,
Comptes Rend us de l' Academie des Sciences de Paris, t. 317, Serie I, p. 975-980, 1993.
12. P. CHARPIN, Tools for coset weight enumerators of some codes, INRIA-report, to appear.
13. P. DELSARTE, Four fundamental pa-rameters of a code and their combinatorial significance,
Informati~n
and Control, vol. 23, N. 5, pp.407-438, 1973.
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