Construction of quasi-cyclic self-dual codes

Abstract

There is a one-to-one correspondence between ℓ-quasi-cyclic codes over a finite field Fq and linear codes over a ring R=Fq[Y]/(Ym−1). Using this correspondence, we prove that every ℓ-quasi-cyclic self-dual code of length mℓ over a finite field Fq can be obtained by the building-up construction, provided that char(Fq)=2 or q≡1(mod4), m is a prime p, and q is a primitive element of Fp. 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 F3 and a new 6-quasi-cyclic self-dual [30,15,10] code over F4. When m=7, we find a new 4-quasi-cyclic self-dual [28,14,9] code over F4 and a new 6-quasi-cyclic self-dual [42,21,12] code over F4.