Constacyclic Codes of Arbitrary Length over Fq+uFq+⋯+ue−1Fq

IEEE Transactions on Information Theory, 2019

Let p be a prime, s be a positive integer, and let R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ∈ R, λ-constacyclic codes of length p s over R are ideals of the quotient ring R[x]/ x p s − λ. In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/ x p s − λ is a chain ring. When R[x]/ x p s − λ is a chain ring, all λ-constacyclic codes of length p s over R are known. In this paper, we establish algebraic structures of all λ-constacyclic codes of length p s over R when R[x]/ x p s − λ is a non-chain ring. We also determine the number of codewords in each of these codes. Using their algebraic structures, we obtain symbol-pair distances, Rosenbloom-Tsfasman (RT) distances, and Rosenbloom-Tsfasman (RT) weight distributions of all constacyclic codes of length p s over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length p s over R is maximumdistance separable (MDS) with respect to the (i) Hamming metric, (ii) symbol-pair metric, and (iii) Rosenbloom-Tsfasman (RT) metric. We also provide an algorithm to decode constacyclic codes of length p s over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair and RT metrics.

Applied Mathematics Letters, 2008

We extend the results of [J.F. Qian, L.N. Zhang, S.X. Zhu, (1 + u)-constacyclic and cyclic codes over F 2 + uF 2 , Appl. Math. Lett. 19 (2006) 820-823. [3]] to codes over the commutative ring R = F p k + uF p k , where p is prime, k ∈ N and u 2 = 0. In particular, we prove that the Gray image of a linear (1 − u)-cyclic code over R of length n is a distance-invariant quasicyclic code of index p k−1 and length p k n over F p k. We also prove that if (n, p) = 1, then every code of length p k n over F p k which is the Gray image of a linear cyclic code of length n over R is permutation-equivalent to a quasicyclic code of index p k−1 .

arXiv (Cornell University), 2012

For λ an n-th power of a unit in a finite chain ring we prove that λ-constacyclic repeated-root codes over some finite chain rings are equivalent to cyclic codes. This allows us to simplify the structure of some constacylic codes. We also study the α+pβconstacyclic codes of length p s over the Galois ring GR(p e , r).

In this study, we consider linear and especially cyclic codes over the non-chain ring $Z_p[v]/<v^p − v>$ where $p$ is a prime. This is a generalization of the case $p = 3.$ Further, in this work the structure of constacyclic codes are studied as well. This study takes advantage mainly from a Gray map which preserves the distance between codes over this ring and $p-$ary codes and moreover this map enlightens the structure of these codes. Furthermore, a MacWilliams type identity is presented together with some illustrative examples.

Finite Fields and Their Applications, 2012

Let F q be a finite field with q = p m elements, where p is an odd prime and m 1. In this paper, we explicitly determine all the μ-constacyclic codes of length 2 n over F q , when the order of μ is a power of 2. We further obtain all the self-dual negacyclic codes of length 2 n over F q and give some illustrative examples. All the repeated-root λ-constacyclic codes of length 2 n p s over F q are also determined for any nonzero λ in F q. As examples all the 2-constacyclic, 3-constacyclic codes of length 2 n 5 s over F 5 and all the 3-constacyclic, 5-constacyclic codes of length 2 n 7 s over F 7 for n 1, s 1 are derived.

2019

Let Zp be the ring of residue classes modulo a prime p. The ZpZp[u, v]-additive cyclic codes of length (α, β) is identify as Zp[u, v][x]-submodule of Zp[x]/〈x α − 1〉 × Zp[u, v][x]/〈x β − 1〉 where Zp[u, v] = Zp + uZp + vZp with u 2 = v2 = uv = vu = 0. In this article, we obtain the complete sets of generator polynomials, minimal generating sets for cyclic codes with length β over Zp[u, v] and ZpZp[u, v]additive cyclic codes with length (α, β) respectively. We show that the Gray image of ZpZp[u, v]-additive cyclic code with length (α, β) is either a QC code of length 4α with index 4 or a generalized QC code of length (α, 3β) over Zp. Moreover, some structural properties like generating polynomials, minimal generating sets of ZpZp[u, v]-additive constacyclic code with length (α, p − 1) are determined.

Journal of Algebra, 2010

For any prime p, all constacyclic codes of length p s over the ring R = F p m + uF p m are considered. The units of the ring R are of the forms γ and α + uβ, where α, β, and γ are nonzero elements of F p m , which provides p m (p m − 1) such constacyclic codes. First, the structure and Hamming distances of all constacyclic codes of length p s over the finite field F p m are obtained; they are used as a tool to establish the structure and Hamming distances of all (α + uβ)-constacyclic codes of length p s over R. We then classify all cyclic codes of length p s over R 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 p s over R is constructed via ring isomorphism, which carries over the results regarding cyclic codes corresponding to γ-constacyclic codes of length p s over R.

Information and Control, 1972

Given an integer m which is a product of distinct primes Pi, a method is given for constructing codes over the ring of integers modulo m from cyclic codes over GF(pi). Specifically, if we are given a cyclic (n, ki) code over GF(pt) with minimum Hamming distance di, for each i, then we construct a code of block length n over the integers modulo m with 1-[~ p~i codewords, which is both linear and cyclic and has minimum Hamming distance mini di. i j k

Finite Fields and Their Applications, 2013

Cyclic, negacyclic and constacyclic codes are part of a larger class of codes called polycyclic codes; namely, those codes which can be viewed as ideals of a factor ring of a polynomial ring. The structure of the ambient ring of polycyclic codes over GR(p a , m) and generating sets for its ideals are considered. It is shown that these generating sets are strong Groebner bases. A method for finding such sets in the case that a = 2 is also given. The Hamming distance of certain constacyclic codes of length ηp s and 2ηp s over Fpm is computed. A method, which determines the Hamming distance of the constacyclic codes of length ηp s and 2ηp s over GR(p a , m), where (η, p) = 1, is described. In particular, the Hamming distance of all cyclic codes of length p s over GR(p 2 , m) and all negacyclic codes of length 2p s over Fpm is determined explicitly.

arXiv (Cornell University), 2023

The Euclidean hull of a linear code C is defined as C ∩ C ⊥ , where C ⊥ denotes the dual of C under the Euclidean inner product. A linear code with zero hull dimension is called a linear complementary dual (LCD) code. A pair (C, D) of linear codes of length n over F q is called a linear complementary pair (LCP) of codes if C ⊕ D = F n q. In this paper, we give a characterization of LCD and LCP of cyclic codes of length q m − 1, m ≥ 1, over the finite field F q in terms of their basic dual zeros and their trace representations. We also formulate the hull dimension of a cyclic code of arbitrary length over F q with respect to its basic dual zero. Moreover, we provide a general formula for the dimension of the intersection of two cyclic codes of arbitrary length over F q based on their basic dual zeros.