A class of three-weight cyclic codes

Abstract

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, a class of threeweight cyclic codes over GF(p) whose duals have two zeros is presented, where p is an odd prime. The weight distribution of this class of cyclic codes is settled. Some of the cyclic codes are optimal. The duals of a subclass of the cyclic codes are also studied and proved to be optimal.

Recent progress on weight distributions of cyclic codes over finite fields

“…It is unnecessary to list all the results on three-weight cyclic codes and we have to omit some results here. In [35,96], the cyclic codes were proved to be three-weight by using quadratic forms. Table 8.…”

Complete weight enumerators of two classes of linear codes

“…For nonbinary linear codes, the (Hamming) weight enumerators, which have been extensively investigated [15], [16], [34], [15], [35], can be obtained from the complete weight enumerators. Further more, the complete weight enumerators are closely related to the deception of some authentication codes constructed from linear codes [11], and used to compute the Walsh transform of monomial functions over finite fields [18].…”

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.…”

“…The cyclic codes over F p with parity-check polynomial h 0 (x)h 1 (x) have been extensively studied in [4,17,20,21]. 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.…”