On matrix-product structure of repeated-root constacyclic codes over finite fields (original) (raw)
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On Repeated-Root Constacyclic Codes of Length 2ampr2^amp^r2ampr over Finite Fields
In this paper we investigate the structure of repeated root constacyclic codes of length 2ampr2^amp^r2ampr over mathbbFps\mathbb{F}_{p^s}mathbbFps with ageq1a\geq1ageq1 and (m,p)=1(m,p)=1(m,p)=1. We characterize the codes in terms of their generator polynomials. This provides simple conditions on the existence of self-dual negacyclic codes. Further, we gave cases where the constacyclic codes are equivalent to cyclic codes.
A class of constacyclic codes over a finite field
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.
Repeated-root constacyclic codes of length 2ps
Finite Fields and Their Applications, 2012
The algebraic structures in term of polynomial generators of all constacyclic codes of length 2p s over the finite field F p m are established. Among other results, all self-dual negacyclic codes of length 2p s , where p ≡ 1 (mod 4) (any m), or p ≡ 3 (mod 4) and m is even, are provided. It is also shown the non-existence of selfdual negacyclic codes of length 2p s , where p ≡ 3 (mod 4), m is odd, and self-dual cyclic codes of length 2p s , for any odd prime p.
A class of repeated-root constacyclic codes over Fpm[u]/〈ue〉 of Type 2
Finite Fields and Their Applications, 2019
Let F p m be a finite field of cardinality p m where p is an odd prime, n be a positive integer satisfying gcd(n, p) = 1, and denote R = F p m [u]/ u e where e ≥ 4 be an even integer. Let δ, α ∈ F × p m. Then the class of (δ + αu 2)-constacyclic codes over R is a significant subclass of constacyclic codes over R of Type 2. For any integer k ≥ 1, an explicit representation and a complete description for all distinct (δ + αu 2)-constacyclic codes over R of length np k and their dual codes are given. Moreover, formulas for the number of codewords in each code and the number of all such codes are provided respectively. In particular, all distinct (δ + αu 2)-contacyclic codes over F p m [u]/ u e of length p k and their dual codes are presented precisely.
Polycyclic codes over Galois rings with applications to repeated-root constacyclic 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.
Structure of repeated-root constacyclic codes of length3psand their duals
Discrete Mathematics, 2013
For any different odd primes ℓ and p, structure of constacyclic codes of length 2ℓ m p n over a finite field Fq of characteritic p and their duals is established in term of their generator polynomials. Among other results, all linear complimentary dual and self-dual constacyclic codes of length 2ℓ m p n over Fq are obtained.
Some Constacyclic Codes over Finite Chain Rings
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).
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.
On a class of repeated-root monomial-like abelian codes
In this paper we study polycyclic codes of length ps1×⋅⋅⋅×psnp^{s_1} × · · · × p^{s_n}ps1×⋅⋅⋅×psn over FpaF_{p^a}Fpa 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.