On repeated-root multivariable codes over a finite chain ring

IEEE Access, 2020

Let R = F p m [u]/ u 3 be the finite commutative chain ring, where p is a prime, m is a positive integer and F p m is the finite field with p m elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over R and their dual codes. We also determine the number of codewords in each repeated-root constacyclic code over R. We also obtain Hamming distances, RT distances, RT weight distributions and ranks (i.e., cardinalities of minimal generating sets) of some repeated-root constacyclic codes over R. Using these results, we also identify some isodual and maximum distance separable (MDS) constacyclic codes over R with respect to the Hamming and RT metrics. INDEX TERMS Cyclic codes, local rings, negacyclic codes, optimal codes.

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: Commutative Algebra, 2017

Let $\mathcal{R}$ be a finite commutative chain ring with nilpotency index 2. In this paper, all constacyclic codes of length $np^{s}$ over $\mathcal{R}$ and their dual codes are determined, where $n,s$ are positive integers with $\gcd(n,p)=1.$ The number of codewords in each constacyclic code of length $np^s$ over $\mathcal{R}$ is also obtained. Besides this, some isodual constacyclic codes of length $np^s$ over $\mathcal{R}$ are listed.

Discrete Mathematics

For any prime number p, positive integers m, k, n satisfying gcd(p, n) = 1 and λ 0 ∈ F × p m , we prove that any λ p k 0-constacyclic code of length p k n over the finite field F p m is monomially equivalent to a matrix-product code of a nested sequence of p k λ 0-constacyclic codes with length n over F p m .

In this paper we study polycyclic codes of length $p^{s_1} × · · · × p^{s_n}$ over $F_{p^a}$ generated by a single monomial. These codes form a special class of abelian codes. We show that these codes arise from the product of certain single variable codes and we determine their minimum Hamming distance. Finally we extend the results of Massey et. al. in [10] on the weight retaining property of monomials in one variable to the weight retaining property of monomials in several variables.

IEEE Access, 2020

Let F 2 m be a finite field of 2 m elements, λ and k be integers satisfying λ, k ≥ 2 and denote For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ + αu 2 )-constacyclic codes over R of length 2 k n, and provide a clear formula to count the number of all these codes. In particular, we conclude that every (δ + αu 2 )-constacyclic code over R of length 2 k n is an ideal generated by at most 2 polynomials in the ring R[x]/ x 2 k n -(δ + αu 2 ) . As an example, we listed all 135 distinct (1 + u 2 )-constacyclic codes of length 4 over F 2 [u] u 4 , and apply our results to determine all 11 self-dual codes among them. INDEX TERMS Type 2 constacyclic code, linear code, repeated-root code, finite chain ring.

IEEE Transactions on Information Theory, 1991

TEMA - Tendências em Matemática Aplicada e Computacional, 2005

In this paper we present a construction technique of cyclic, BCH, alternat, Goppa and Srivastava codes over a local finite commutative rings with identity.

Discrete Mathematics, 1997

It is well known that cyclic linear codes of length n over a (finite) field F can be characterized in terms of the factors of the polynomial x"-1 in F[x]. This paper investigates cyclic linear codes over arbitrary (not necessarily commutative) finite tings and proves the above characterization to be true for a large class of such codes over these rings. (~

Advances in Mathematics of Communications, 2019

Galois images of polycyclic codes over a finite chain ring S and their annihilator dual are investigated. The case when a polycyclic codes is Galois-disjoint over the ring S, is characterized and, the trace codes and restrictions of free polycyclic codes over S are also determined givind an analogue of Delsarte theorem among trace map, any S-linear code and its annihilator dual.