TY - JOUR

T1 - Construction of quasi-cyclic self-dual codes

AU - Han, Sunghyu

AU - Kim, Jon Lark

AU - Lee, Heisook

AU - Lee, Yoonjin

N1 - Funding Information:
E-mail addresses: sunghyu@kut.ac.kr (S. Han), jl.kim@louisville.edu (J.-L. Kim), hsllee@ewha.ac.kr (H. Lee), yoonjinl@ewha.ac.kr (Y. Lee). 1 S. Han was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF), which is supported by the Ministry of Education, Science and Technology (2010-0007232). 2 Y. Lee is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2011-0003516).

PY - 2012/5

Y1 - 2012/5

N2 - There is a one-to-one correspondence between ℓ-quasi-cyclic codes over a finite field F q and linear codes over a ring R=F q[Y]/(Y m-1). Using this correspondence, we prove that every ℓ-quasi-cyclic self-dual code of length mℓ over a finite field F q can be obtained by the building-up construction, provided that char(F q)=2 or q=1(mod4), m is a prime p, and q is a primitive element of F p. We determine possible weight enumerators of a binary ℓ-quasi-cyclic self-dual code of length pℓ (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ℓ-quasi-cyclic codes of length 3ℓ) optimal self-dual codes of lengths 30,36,42,48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m=5, we obtain a new 8-quasi-cyclic self-dual [40,20,12] code over F 3 and a new 6-quasi-cyclic self-dual [30,15,10] code over F 4. When m=7, we find a new 4-quasi-cyclic self-dual [28,14,9] code over F 4 and a new 6-quasi-cyclic self-dual [42,21,12] code over F 4.

AB - There is a one-to-one correspondence between ℓ-quasi-cyclic codes over a finite field F q and linear codes over a ring R=F q[Y]/(Y m-1). Using this correspondence, we prove that every ℓ-quasi-cyclic self-dual code of length mℓ over a finite field F q can be obtained by the building-up construction, provided that char(F q)=2 or q=1(mod4), m is a prime p, and q is a primitive element of F p. We determine possible weight enumerators of a binary ℓ-quasi-cyclic self-dual code of length pℓ (with p a prime) in terms of divisibility by p. We improve the result of Bonnecaze et al. (2003) [3] by constructing new binary cubic (i.e., ℓ-quasi-cyclic codes of length 3ℓ) optimal self-dual codes of lengths 30,36,42,48 (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When m=5, we obtain a new 8-quasi-cyclic self-dual [40,20,12] code over F 3 and a new 6-quasi-cyclic self-dual [30,15,10] code over F 4. When m=7, we find a new 4-quasi-cyclic self-dual [28,14,9] code over F 4 and a new 6-quasi-cyclic self-dual [42,21,12] code over F 4.

KW - Building-up construction

KW - Cubic code

KW - Quasi-cyclic self-dual code

KW - Quintic code

UR - http://www.scopus.com/inward/record.url?scp=84862832226&partnerID=8YFLogxK

U2 - 10.1016/j.ffa.2011.12.006

DO - 10.1016/j.ffa.2011.12.006

M3 - Article

AN - SCOPUS:84862832226

VL - 18

SP - 613

EP - 633

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

SN - 1071-5797

IS - 3

ER -