Extremal quasi-cyclic self-dual codes over finite fields

We study self-dual codes over a factor ring R = F-q[X]/(X-m - 1) of length l, equivalently, tquasi-cyclic self-dual codes of length ml over a finite field F-q, provided that the polynomial X-m - 1 has exactly three distinct irreducible factors in F-q [X], where F-q is the finite field of order q. There are two types of the ring R depending on how the conjugation map acts on the minimal ideals of R. We show that every self dual code over the ring R. of the first type with length >= 6 has free rank >= 2. This implies that every l-quasi-cyclic self dual code of length ml >= 6m over F-q can be obtained by the building-up construction, where m corresponds to the ring R. of the first type. On the other hand, there exists a self-dual code of free rank <= 1 over the ring R. of the second type. We explicitly determine the forms of generator matrices of all self-dual codes over R of free rank <= 1. For the case that in = 7, we find 9828 binary l-quasi-cyclic self-dual codes of length 70 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring R. of the second type. These codes are all new codes. Furthermore, for the case that m = 17, we find 1566 binary 4-quasi-cyclic self-dual codes of length 68 with minimum weight 12, up to equivalence, which are constructed from self-dual codes over the ring 1 of the first type. (C) 2018 Elsevier Inc. All rights reserved.