The weight distributions of a class of cyclic codes

Abstract

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in [14,15,16]. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method.2000 Mathematics Subject Classification. 94B15,11T71,11T24.

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

“…Therefore, a reducible cyclic code is, basically, the direct sum of these irreducible cyclic codes. Reducible cyclic codes, whose parity-check polynomials are factorizable in exactly two different irreducible factors have been extensively studied (see, for example, [4], [15], [11], [8], [3], [14], [16] and [12]). Now, each one of these two irreducible factors might or might not correspond to the parity-check polynomial of a one-weight irreducible cyclic code.…”

“…Most of the recent efforts along this line of research have been focused on the study of reducible cyclic codes constructed as the direct sum of two two-weight irreducible cyclic codes. In fact, through the easy-to-apply characterization for all semiprimitive two-weight irreducible cyclic codes over any finite field, that was recently presented in [13] is interesting to note that most of the families of reducible cyclic codes studied in [8], [3], [14], [16] and [12], are constructed as a direct sum of two different semiprimitive two-weight irreducible cyclic codes of the same dimension. In this paper, we present a new class of reducible cyclic codes constructed as the direct sum of two one-weight irreducible cyclic codes.…”

Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

“…In the past decades, the weight distributions of cyclic codes have been studied extensively. Interested readers may refer to [1][2][3]12,23,24,28,29,32,33,35,37] and the survey paper [6] for irreducible cyclic codes, and to [8][9][10][11]13,[19][20][21][22]25,26,34,36,[38][39][40]43] and the references therein for cyclic codes with two or three zeroes. However, due to increased difficulties, there are very few results for cyclic codes with more than three zeroes.…”

The weight distributions of a class of non-primitive cyclic codes with two nonzeros

“…Some cases of this class of cyclic codes have been well studied in [4,11,[16][17][18][19][20]. Let m be a positive integer, r = q m and α be a primitive element of F r .…”