Determining the full weight distribution of any irreducible cyclic code over any finite field of dimension two

Abstract

In coding theory, a very interesting problem (but at the same time, a very difficult one) is to determine the weight distribution of a given code. This problem is even more interesting for cyclic codes, and this is so, mainly because they possess a rich algebraic structure, which can be utilized in a variety of ways. For example, in Engineering Telecommunications, this structure is commonly employed to design very efficient coding and decoding algorithms. In this note, we are going to use a characterization of all semiprimitive two-weight irreducible cyclic codes over any finite field, recently presented in [8], in order to determine the full weight distribution of any irreducible cyclic code over any finite field of dimension two. In fact, the relevance of our results is that by means of them we can actually directly determine the full weight distribution of any irreducible cyclic code, of dimension one or two, just by knowing its length.