Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8895639 | Finite Fields and Their Applications | 2018 | 18 Pages |
Abstract
We study self-dual codes over a factor ring R=Fq[X]/(Xmâ1) of length â, equivalently, â-quasi-cyclic self-dual codes of length mâ over a finite field Fq, provided that the polynomial Xmâ1 has exactly three distinct irreducible factors in Fq[X], where Fq 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 â¥6 has free rank â¥2. This implies that every â-quasi-cyclic self-dual code of length mââ¥6m over Fq 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 â¤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 â¤1. For the case that m=7, we find 9828 binary 10-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 R of the first type.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Hyun Jin Kim, Yoonjin Lee,