A class of six-weight cyclic codes and their weight distribution

Abstract

In this paper, a family of reducible cyclic codes over F p whose duals have four zeros is presented, where p is an odd prime. Furthermore, the weight distribution of these cyclic codes is determined.

Two classes of cyclic codes and their weight enumerator

“…Remark C m,d,2k−1 in the case of k = 2 has been studied in [15], and the minimum distance has different expression between cases of k = 2 and 3 ≤ k ≤ m+1 2 , therefore, only the case of 3 ≤ k ≤ m+1 2 is presented here.…”

“…Zhou and Ding [29] proved that the cyclic codes over F p with parity-check polynomial h −0 (x)h 1 (x) have three nonzero weights, and determined their weight distributions. In [15], it was proved that the cyclic codes over F p with parity-check polynomial h 0 (x)h −0 (x)h 1 (x) have six nonzero weights and their weight distributions were determined as well. General cases are more interesting.…”

A class of cyclic codes whose duals have five zeros

“…For information on the weight distribution of irreducible cyclic codes, the reader is referred to [2,3,5,6,13]. Information on the weight distribution of reducible cyclic codes could be found in [4,[7][8][9][10][11][12][15][16][17][18][19][21][22][23][24]. In this paper, we will determine the weight distribution of a class of reducible cyclic codes whose duals have five zeros.…”

“…In [23], Zhengchun Zhou and Cunsheng Ding proved the cyclic codes over F p with parity-check polynomial h −0 (x)h 1 (x) have three nonzero weights and determined their weight distribution. And in [16], the authors proved the cyclic codes over F p with parity-check polynomial h 0 (x)h −0 (x)h 1 (x) have six nonzero weights and determined their weight distribution. Let C ( p,m,k) be the cyclic code with parity-check polynomial…”

A Family of Six-Weight Reducible Cyclic Codes and their Weight Distribution

“…In fact, for the codes in this new class we will explicitly give their full weight distribution, and show that none of them can be projective. Recently, on the other hand, a family of six-weight reducible cyclic codes and their weight distribution, was presented in [7], however it is worth noting that, unlike what is done here, the codes in such family are constructed as the direct sum of three different irreducible cyclic codes of the same dimension.…”