Multi-twisted codes over finite fields and their dual codes

Abstract

Let Fq denote the finite field of order q, let m1,m2,⋯,mℓ be positive integers satisfying gcd⁡(mi,q)=1 for 1≤i≤ℓ, and let n=m1+m2+⋯+mℓ. Let Λ=(λ1,λ2,⋯,λℓ) be fixed, where λ1,λ2,⋯,λℓ are non-zero elements of Fq. In this paper, we study the algebraic structure of Λ-multi-twisted codes of length n over Fq and their dual codes with respect to the standard inner product on Fqn. We provide necessary and sufficient conditions for the existence of a self-dual Λ-multi-twisted code of length n over Fq, and obtain enumeration formulae for all self-dual and self-orthogonal Λ-multi-twisted codes of length n over Fq. We also derive some sufficient conditions under which a Λ-multi-twisted code is linear with complementary dual (LCD). We determine the parity-check polynomial of all Λ-multi-twisted codes of length n over Fq and obtain a BCH type bound on their minimum Hamming distances. We also determine generating sets of dual codes of some Λ-multi-twisted codes of length n over Fq from the generating sets of the codes. Besides this, we provide a trace description for all Λ-multi-twisted codes of length n over Fq by viewing these codes as direct sums of certain concatenated codes, which leads to a method to construct these codes. We also obtain a lower bound on their minimum Hamming distances using their multilevel concatenated structure.