Problems of Information Transmission, Vol. 41, No. 2, 2005, pp. 130–133. Translated from Problemy Peredachi Informatsii, No. 2, 2005, pp. 68–71.
Original Russian Text Copyright
2005 by Mullayeva.
Interrelation of the Cyclic and Weight Structure
of Codes over GF (q) and Proportionality Classes
I. I. Mullayeva
Center of Economic Reforms, Baku, Azerbaijan Republic
Received August 18, 2003; in ﬁnal form, November 16, 2004
Abstract—To determine the weight structure of cyclic codes, we establish an interrelation of
the cyclic structure of a code and classes of proportional elements.
The proportionality relation of elements of the algebra A
of polynomials in variable x over the
ﬁeld GF(q), q>2, modulo the polynomial x
−1 is an equivalence relation ; therefore, the algebra
is split into pairwise disjoint classes of proportional elements. If α
, ..., α
is the multi-
plicative group GF(q)
of GF (q), then the proportionality class P
,wherez(x) ∈ A
, z(x) =0,
consists of the q − 1elementsα
z(x), ..., α
z(x); i.e., A
− 1)/(q − 1) dis-
tinct nonzero classes. Clearly, elements of one class have the same period , as well as the same
support set and Hamming weight , and are characterized by their unique normalized polynomial.
Consider an ideal J ∈ A
, i.e., a cyclic code over GF (q) (hereafter we assume that q>2and
(n, q) = 1) with generator polynomial g(x)=(x
− 1)/h(x),whereh(x) is a check polynomial
of the code of degree m and order n =ord(h(x)). The ideal consists of cycles (i.e., cyclic shifts of
one and the same polynomial z(x)), whose periods are various divisors of n [2,5]; also, the ideal as
a subspace of A
− 1)/(q − 1) proportionality classes. Clearly, the existence of two
diﬀerent partitions of an ideal presumes certain dependence between proportionality classes and
cycles of the ideal.
An ideal J ∈ A
is a direct sum of minimal ideals [3, 6]
then the polynomial h(x) has the following form over GF (q):
(x) is an irreducible check polynomial of the ideal J
of degree m
and order n
1 ≤ i ≤ t,andn =lcm(n
) is the order of h(x).
An element of an ideal J is characterized by its unique minimal polynomial [6,7]. Denote by C
the set of all elements of J with the minimal polynomial c(x)oftheform
(x), 1 ≤ k ≤ t. (3)
All elements of C have period n
| n. Since the set C is closed with respect to
multiplication by elements of GF (q) and cyclic shifts , counting the number of elements in C in
2005 Pleiades Publishing, Inc.