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We recall a classic lower bound on the minimum Hamming distance of constacyclic codes over finite fields, analogous to the well-known BCH bound for cyclic codes. This BCH-like bound serves as a foundation for proposing some minimum-distance lower bounds for single-generator quasi-twisted (QT) codes. Associating each QT code with a constacyclic code over an extension field, we obtain the first bound. This is the QT analogue to a result in the literature for quasi-cyclic codes. We point out some weaknesses in this bound and propose a novel bound that takes into account the Chinese remainder theorem approach to QT codes as well as the BCH bound of constacyclic codes. This proposed bound, in contrast to previous bounds in the literature, does not presuppose a specific form of code generator and does not require calculations in any extension field. We illustrate that our bound meets the one in the literature when the code generator adheres to the specific form assumed in that study. Various numerical examples enable us to compare and discuss these bounds. [ABSTRACT FROM AUTHOR]

Let p ≠ 3 be any prime. The algebraic structures of all λ -constacyclic codes of length 3 p s over the finite commutative chain ring R = F p m + u F p m are provided by classifying all ideals of the local ring R [ x ] 〈 x 3 p s − λ 〉. Using these structures, in this paper, the exact values of Hamming distances of all such λ -constacyclic codes of length 3 p s are established. As an application, we identify all maximal distance separable λ -constacyclic codes of length 3 p s. [ABSTRACT FROM AUTHOR]

Codes over Galois rings have been studied extensively during the last three decades. Negacyclic codes over \mathop \mathrm GR\nolimits (2^a,m) of length 2^s have been characterized: the ring \mathcal R2(a,m,-1)= {\mathrm{ GR}}(2^{a},m)[x]/\langle x^{2^{s}}+1\rangle is a chain ring. Furthermore, these results have been generalized to \lambda -constacyclic codes for any unit \lambda of the form 4z-1 , . In this paper, we study more general cases and investigate all cases, where {\mathcal{ R}}_{p}(a,m,\gamma) = {\mathrm{ GR}}(p^{a},m)[x]/\langle x^{p^{s}}-\gamma \rangle is a chain ring. In particular, the necessary and sufficient conditions for the ring {\mathcal{ R}}_{p}(a,m,\gamma) to be a chain ring are obtained. In addition, by using this structure we investigate all \gamma -constacyclic codes over \mathop {\mathrm {GR}}\nolimits (p^{a},m) when {\mathcal{ R}}_{p}(a,m,\gamma)$ is a chain ring. The necessary and sufficient conditions for the existence of self-orthogonal and self-dual \gamma$ -constacyclic codes are also provided. Among others, for any prime p$ , the structure of is used to establish the Hamming and homogeneous distances of $\gamma$ -constacyclic codes. [ABSTRACT FROM PUBLISHER]

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]

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]

We study all constacyclic codes of length 28 over GR(R, m), the Galois extension ring of dimension m of the ring R = F2 + uF2. The units of the ring GR(R, m) are of the forms α, and α + uβ, where α, β are nonzero elements of F2m, which correspond to 2m (2m - 1) such constacyclic codes. First, the structure and Hamming distances of (1 + uγ)-constacydlic codes are established. We then classify all cyclic codes of length 28 over GR(R, m), and obtain a formula for the number of those cyclic codes, as well as the number of codewords in each code. Finally, one-to-one correspondences between cyclic and α-constacyclic codes, as well as (1 + uγ)-constacyclic and (α + uβ)-constacyclic codes are provided via ring isomorphisms, that allow us to carry over the results about cyclic and (1 + uγ)-constacyclic accordingly to all constacyclic codes of length 28 over GR(R, m). [ABSTRACT FROM AUTHOR]

Various kinds of distances of all negacyclic codes of length 2s over are completely determined. Using our structure theorems of negacyclic codes of length 2s over Z2a, we first calculate the Hamming distances of all such negacyclic codes, which particularly lead to the Hamming weight distributions and Hamming weight enumerators of several codes. These Hamming distances are then used to obtain their homogeneous, Lee, and Euclidean distances. Our techniques are extendable to the more general class of constacyclic codes, namely, the λ-constacyclic codes of length 2s over Z2a, where λ is any unit of Z2a with the form 4k - 1. We establish the Hamming, homogeneous, Lee, and Euclidean distances of all such constacyclic codes. [ABSTRACT FROM AUTHOR]