TY - JOUR
T1 - Extremal quasi-cyclic self-dual codes over finite fields
AU - Kim, Hyun Jin
AU - Lee, Yoonjin
N1 - Funding Information:
The author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827) and also by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1A2B2004574).
Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2018/7
Y1 - 2018/7
N2 - We study self-dual codes over a factor ring R=Fq[X]/(Xm−1) of length ℓ, equivalently, ℓ-quasi-cyclic self-dual codes of length mℓ over a finite field Fq, provided that the polynomial Xm−1 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 ≥6 has free rank ≥2. This implies that every ℓ-quasi-cyclic self-dual code of length mℓ≥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 ≤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 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.
AB - We study self-dual codes over a factor ring R=Fq[X]/(Xm−1) of length ℓ, equivalently, ℓ-quasi-cyclic self-dual codes of length mℓ over a finite field Fq, provided that the polynomial Xm−1 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 ≥6 has free rank ≥2. This implies that every ℓ-quasi-cyclic self-dual code of length mℓ≥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 ≤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 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.
KW - Extremal code
KW - Finite field
KW - Quasi-cyclic code
KW - Self-dual code
UR - http://www.scopus.com/inward/record.url?scp=85046665416&partnerID=8YFLogxK
U2 - 10.1016/j.ffa.2018.04.013
DO - 10.1016/j.ffa.2018.04.013
M3 - Article
AN - SCOPUS:85046665416
SN - 1071-5797
VL - 52
SP - 301
EP - 318
JO - Finite Fields and their Applications
JF - Finite Fields and their Applications
ER -