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In this paper, we introduce a new class of linear codes over finite fields, viz. skew multi-twisted codes (or skew generalized quasi-twisted codes), which is a generalization of some well-known classes of linear codes such as cyclic codes and quasi-cyclic codes. We also study algebraic structures of skew multi-twisted codes and their Galois duals (i.e., orthogonal complements with respect to the Galois inner product). We also view skew multi-twisted codes as direct sums of certain concatenated codes, which gives rise to a method to construct these codes. We also determine a lower bound on their minimum Hamming distances using their multilevel concatenated structure. Apart from this, we determine the parity-check polynomial of each skew multi-twisted code, and obtain BCH type bounds on their minimum Hamming distances. We also determine generating sets of Galois duals of some skew multi-twisted codes from generating sets of these codes. We also derive necessary and sufficient conditions under which a skew multi-twisted code is (i) Galois self-dual, (ii) Galois self-orthogonal and (iii) Galois LCD (linear with complementary dual). As an application of this result, we also provide enumeration formulae for all Hermitian self-dual, Hermitian self-orthogonal, Hermitian LCD and Euclidean LCD multi-twisted codes over finite fields. We also obtain many linear codes with best known and optimal parameters from 1-generator skew multi-twisted codes over finite fields F 8 and F 9. [ABSTRACT FROM AUTHOR]

The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming distance is gained, and it is possible to design efficient decoding algorithms. In this paper, we give a version of the Hartmann–Tzeng bound that works for a wide class of skew cyclic codes. We also provide a practical method for constructing them with designed distance. For skew BCH codes, which are covered by our constructions, we discuss decoding algorithms. Detailed examples illustrate both the theory as the constructive methods it supports. [ABSTRACT FROM AUTHOR]

The structure of λ -constacyclic codes of length 2 s over the Galois ring GR ( 2 a , m ) is obtained, for any unit λ of the form 4 z − 1 , z ∈ GR ( 2 a , m ) . The duals codes and necessary and sufficient conditions for the existence of a self-dual λ -constacyclic code are provided. Among others, this structure is used to establish the Hamming, homogeneous, and Rosenbloom–Tsfasman distances, and Rosenbloom–Tsfasman weight distribution of all such constacyclic codes. [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be any prime and l ≠ 3 be any odd prime with gcd ⁡ ( p , l ) = 1 . The multiplicative group F q ⁎ = 〈 ξ 〉 can be decomposed into mutually disjoint union of gcd ⁡ ( q − 1 , 3 l p s ) cosets over the subgroup 〈 ξ 3 l p s 〉 , where ξ is a primitive ( q − 1 ) th root of unity. We classify all repeated-root constacyclic codes of length 3 l p s over the finite field F q into some equivalence classes by this decomposition, where q = p m , s and m are positive integers. According to these equivalence classes, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length 3 l p s over F q and their dual codes. Self-dual cyclic codes of length 3 l p s over F q exist only when p = 2 . We give all self-dual cyclic codes of length 3 ⋅ 2 s l over F 2 m and their enumeration. We also determine the minimum Hamming distance of these codes when gcd ⁡ ( 3 , q − 1 ) = 1 and l = 1 . [ABSTRACT FROM AUTHOR]

Let F q be a finite field of q elements with q ≡ 3 ( mod 8 ) and p | q − 1 , where p is an odd prime. In this paper, we use matrix method to give all primitive idempotents in two rings F q [ x ] / 〈 x 4 p n − 1 〉 and F q [ x ] / 〈 x 8 p n − 1 〉 . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length 4 p n and 8 p n over F q . [ABSTRACT FROM AUTHOR]

Let p be a prime number. In this paper, we study cyclic codes over the ring Z p [ u , v ] / 〈 u 2 , v 2 , u v − v u 〉 . We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes. We obtain all except one ternary optimal code of length 12 as the Gray image of the cyclic codes over the ring Z p [ u , v ] / 〈 u 2 , v 2 , u v − v u 〉 . [ABSTRACT FROM AUTHOR]

Let F q be a finite field with q elements and n = l 1 m 1 l 2 m 2 , m 1 ≥ 1 , m 2 ≥ 1 , where l 1 , l 2 are distinct primes and l 1 l 2 | q − 1 . In this paper, we give all irreducible factors of x l 1 m 1 l 2 m 2 − 1 over F q and all primitive idempotents in a ring F q [ x ] / 〈 x l 1 m 1 l 2 m 2 − 1 〉 . Furthermore, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length l 1 m 1 l 2 m 2 over F q . [ABSTRACT FROM AUTHOR]

Abstract: For any prime p, all constacyclic codes of length over the ring are considered. The units of the ring are of the forms γ and , where , and γ are nonzero elements of , which provides such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length over the finite field are obtained; they are used as a tool to establish the structure and Hamming distances of all -constacyclic codes of length over . We then classify all cyclic codes of length over and obtain the number of codewords in each of those cyclic codes. Finally, a one-to-one correspondence between cyclic and γ-constacyclic codes of length over is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γ-constacyclic codes of length over . [Copyright &y& Elsevier]

Abstract: We investigate negacyclic and cyclic codes of length over the finite field . Negacyclic codes of length are precisely the ideals of the chain ring . This structure is then used to obtain the Hamming distance distribution of the class of such negacyclic codes, which also provides Hamming weight distributions and enumerations of several codes. An one-to-one correspondence between negacyclic and cyclic codes is established to carry accordingly those results of negacyclic codes to cyclic codes. [Copyright &y& Elsevier]