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

Reducible cyclic codes with exactly two nonzero weights were first studied by T. Helleseth [4] and J. Wolfmann [15]. Later on, G. Vega [11], set forth the sufficient numerical conditions in order that a cyclic code, constructed as the direct sum of two one-weight cyclic codes, has exactly two nonzero weights, and conjectured that there are no other reducible two-weight cyclic codes of this type. In this paper we present a new class of cyclic codes constructed as the direct sum of two one-weight cyclic codes. As will be shown, this new class of cyclic codes is in accordance with the previous conjecture, since its codes have exactly six nonzero weights. In fact, for these codes, we will also give their full weight distribution, showing that none of them can be projective. On the other hand, recently, a family of six-weight reducible cyclic codes and their weight distribution, was presented by Y. Liu, et al. [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 irreducible cyclic codes.