The complete weight distribution of a family of irreducible cyclic codes of dimension two

An important family of codes for data storage systems, cryptography, consumer electronics, and network coding for error control in digital communications are the so-called cyclic codes. This kind of linear codes are also important due to their efficient encoding and decoding algorithms. Because of this, cyclic codes have been studied for many years, however their complete weight distributions are known only for a few cases. The complete weight distribution has a wide range of applications in many research fields as the information it contains is of vital use in practical applications. Unfortunately, obtaining these distributions is in general a very hard problem that normally involves the evaluation of sophisticated exponential sums, which leaves this problem open for most of the cyclic codes. In this paper we determine, for any finite field $\bbbf_q$, the explicit factorization of any polynomial of the form $x^{q+1}-c$, where $c \in \bbbf_{q}^*$. Then we use this result to obtain, without the need to evaluate any kind of exponential sum, the complete weight distributions of a family of irreducible cyclic codes of dimension two over any finite field. As an application of our findings, we employ the complete weight distributions of some irreducible cyclic codes presented here to construct systematic authentication codes, showing that they are optimal or almost optimal.