Construction of quasi-cyclic self-dual codes over finite fields

Our goal of this paper is to find a construction of all l-quasi-cyclic self-dual codes over a finite field F-q of length ml for every positive even integer l. In this paper, we study the case where x(m) - 1 has an arbitrary number of irreducible factors in F-q[x]; in the previous studies, only some special cases where x(m) - 1 has exactly two or three irreducible factors in F-q[x], were studied. Firstly, the binary code case is completed: for any even positive integer l, every binary l-quasi cyclic self-dual code can be obtained by our construction. Secondly, we work on the q-ary code cases for an odd prime power q. We find an explicit method for construction of all l-quasi-cyclic self-dual codes over F-q of length ml for any even positive integer t, where we require that q equivalent to 1 (mod 4) if the index t >= 6. By implementation of our method, we obtain a new optimal binary self-dual code [172, 86, 24], which is also a quasi-cyclic code of index 4.