Multi-twisted codes over finite fields and their dual codes

Abstract

Let Fq denote the finite field of order q, let m1, m2, · · · , m ℓ be positive integers satisfying gcd(mi, q) = 1 for 1 ≤ i ≤ ℓ, and let n = m1 + m2 + · · · + m ℓ . Let Λ = (λ1, λ2, · · · , λ ℓ ) be fixed, where λ1, λ2, · · · , λ ℓ 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 F n q . 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 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.Theorem and the results derived in Ling and Solé [12]. They also obtained an improved lower bound on their minimum Hamming distances. Güneri et al. [6] decomposed GQC codes as direct sums of concatenated codes, which leads to a trace formula and a minimum distance bound for GQC codes. Jia [9] decomposed quasi-twisted (QT) codes and their dual codes over finite fields to a direct sum of linear codes over rings, and provided a method to construct quasi-twisted codes by using generalized discrete Fourier transform. Saleh and Esmaeili [16] gave some sufficient conditions under which a quasi-twisted code is LCD. In a recent work, Aydin and Halilovic [1] introduced multi-twisted (MT) codes as generalization of quasi-twisted codes. They studied basic properties of 1-generator multi-twisted codes and provided a lower bound on their minimum Hamming distances. The family of multi-twisted codes is much broader as compared to quasi-twisted and constacyclic codes.Throughout this paper, let F q denote the finite field of order q, and let Λ = (λ 1 , λ 2 , · · · , λ ℓ ) be fixed, where λ 1 , λ 2 , · · · , λ ℓ are non-zero elements of F q . Let n = m 1 + m 2 + · · ·+ m ℓ , where m 1 , m 2 , · · · , m ℓ are positive integers coprime to q. The main aim of this paper is to study the algebraic structure of Λ-multi-twisted codes of length n over F q and their dual codes with respect to the standard inner product on F n q . Enumeration formulae for their two interesting subclasses, viz. self-dual and self-orthogonal codes, are also obtained. These enumeration formulae are useful in the determination of complete lists of inequivalent self-dual and self-orthogonal Λ-multitwisted codes [7, Section 9.6...

Linear Complementary Pairs of Multi-twisted Codes and their Characterizations

“…As f j (x) is irreducible, we have F j is a field. As (m i , p) = 1, from [9] we have M i = r j=1 l ji F j , where l ji F j = (0) if l ji = 0 and l ji F j = F j otherwise. Hence, we have…”

“…They have shown that there are codes with better parameters in this class compared to the other known linear codes. Later, Sharma et al [9] described the algebraic structure of multi-twisted codes and its dual codes. They have obtained some conditions under which the multi-twisted code is a linear complementary dual (LCD).…”

Every Fq -linear code is equivalent to a non-trivial multi-twisted code

“…Cyclic and constacyclic linear codes have been generalized in various forms. Quasi-cyclic (QC) codes, quasi-twisted (QT) codes, generalized quasi-cyclic (GQC) codes, multi-twisted (MT) codes are important generalizations among cyclic and constacyclic codes which have attracted many researchers; see [14], [3,4,8], [2,6,13] and [1,10] respectively.…”

On Z₂Z₂[u³]-Additive Cyclic and Complementary Dual Codes

“…For a linear code C , if C ∩ C ⊥ = {0}, then we called it an LCD code. LCD codes over finite fields were mainly studied in [20]- [29]. Shi et al studied LCD codes over Galois rings, and obtained some classes of asymptotically good LCD codes [30].…”