Repeated-root Constacyclic Codes Research Articles

On constacyclic codes of length 9ps over 𝔽pm and their optimal codes

The problem of classifying constacyclic codes over a finite field, both the Hamming distance and the algebraic structure, is an interesting problem in algebraic coding theory. For the repeated-root constacyclic codes of length [Formula: see text] over [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text] does not divide [Formula: see text], the problem has been solved completely for all [Formula: see text] and partially for [Formula: see text]. In this paper, we solve the problem for [Formula: see text] and all primes [Formula: see text] different from [Formula: see text] and [Formula: see text]. In particular, we characterize the Hamming distance of all repeated-root constacyclic codes of length [Formula: see text] over [Formula: see text]. As an application, we identify all optimal and near-optimal codes with respect to the Singleton bound of these types, namely, MDS, almost-MDS, and near-MDS codes.

Repeated-root constacyclic codes of length $ p_1p_2^t p^s $ and their dual codes

<abstract><p>Let $ \mathbb{F}_q $ be the finite field with $ q = p^{k} $ elements, and $ p_{1}, p_{2} $ be two distinct prime numbers different from $ p $. In this paper, we first calculate all the $ q $-cyclotomic cosets modulo $ p_1p_2^t $ as a preparation for the following parts. Then we give the explicit generator polynomials of all the constacyclic codes of length $ p_1p_2^tp^s $ over $ \mathbb{F}_q $ and their dual codes. In the rest of this paper, we determine all self-dual cyclic codes of length $ p_1p_2^t p^s $ and their enumeration. This answers a question recently asked by B. Chen, H.Q.Dinh and Liu. In the last section, we calculate the case of length $ 5\ell p^{s} $ as an example.</p></abstract>

Symbol-Triple Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over Fq+uFq+u2Fq

Let p be an odd prime, where ϑ and m are positive integers. Let ψ be a nonzero element of the finite field Fq, where q=pm, and R=Fq+uFq+u2Fq(u3=0). In this paper, we determine completely the symbol-triple distances of all ψ-constacyclic codes of length pϑ over R.

On the symbol-pair distance of some classes of repeated-root constacyclic codes over Galois ring

Let $$\gamma = 4z-1$$ be an unit of Type $$(*^{-})$$ of the Galois ring $${{\,\mathrm{GR}\,}}(2^a, m)$$ . The $$\gamma$$ -constacyclic codes of length $$2^s$$ over the Galois ring $${{\,\mathrm{GR}\,}}(2^a, m)$$ are precisely the ideals $$\langle (x +1)^i \rangle$$ , $$0 \le i \le 2^sa$$ of the chain ring $$\mathfrak {R}(a,m, \gamma ) = \dfrac{{{\,\mathrm{GR}\,}}(2^a,m)[x]}{\langle {x^{2^s}} - \gamma \rangle }$$ . This structure is used to determine the symbol pair distance of $$\gamma$$ -constacyclic codes of length $$2^s$$ over $${{\,\mathrm{GR}\,}}(2^a, m)$$ . The exact symbol-pair distances for all such $$\gamma$$ -constacyclic codes of length $$2^s$$ over $${{\,\mathrm{GR}\,}}(2^a, m)$$ are obtained. Also, we provide the MDS symbol-pair codes of length $$2^s$$ over $${{\,\mathrm{GR}\,}}(2^a, m)$$ and some examples are computed.

Repeated-Root Constacyclic Codes With Pair-Metric

Symbol-pair codes are block codes when considering the pair-metric, and are designed to protect against pair-errors occur in the pair-read vector which consisting of overlapping pairs of symbols. A code that achieves the analog of Singleton bound for symbol-pair codes is called a maximum distance separable (MDS) symbol-pair code. The contribution of this letter is twofold. First, we give a lower bound of minimum pair-distance of repeated-root constacyclic codes in terms of minimum Hamming distance, which extends the lower bound given by Chen, Lin and Liu. Second, we give three new classes of MDS symbol-pair codes. As we know, for an MDS symbol-pair code with a fixed minimum pair-distance, the larger the code length of the code has, the higher the code rate is. The first class of MDS symbol-pair codes we show has unbounded code length and the other two classes of MDS symbol-pair codes have larger code lengths than the code lengths of existing MDS symbol-pair codes under the same minimum pair-distance 6.

Repeated-Root Constacyclic Codes Over the Chain Ring Fpm[u]/⟨u3⟩

Let Z = F p m [u]/(u 3 ) be the finite commutative chain ring, where p is a prime, m is a positive integer and Fpm is the finite field with pm elements. In this paper, we determine all repeated-root constacyclic codes of arbitrary lengths over Z and their dual codes. We also determine the number of codewords in each repeated-root constacyclic code over Z. 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 Z. Using these results, we also identify some isodual and maximum distance separable (MDS) constacyclic codes over Z with respect to the Hamming and RT metrics.

B-Symbol Distance of Constacylic Codes of Length ps Over Fp m + uFp m

In this research paper, the repeated-root constacyclic codes over the chain ring F p m + uF p m are considered, where p is a prime and m > 0 is any integer. The b-symbol distance for prime power length, i.e. p s is also studied for any integer s > 0. The Hamming and symbol-pair distances of all δ-constacyclic codes have been thoroughly studied in [18], where δ is an unit in the ring F p m + uF p m which is of the form ζ and φ + uφ, where 0 ≠ 6 φ, φ, ζ ∈ F p m. In this paper, the b-symbol distance of all such δ-constacyclic codes of prime power length is computed for 1 ≤ b ≤ ⌊p/2⌋. Furthermore, as an application, all MDS b-symbol constacyclic codes of length p s over F p m + uF p m are established.

On the Hamming Distances of Constacyclic Codes of Length 5pS

Let p be a prime, s, m be positive integers, and λ be a nonzero element of the finite field F p(m) . In this paper, the algebraic structures of constacyclic codes of length 5 p s (p ≠ 5) are obtained, which provide all self-dual, self-orthogonal and dual containing codes. Moreover, the exact values of the Hamming distances of all such codes are completely determined. Among other results, we obtain the degrees of the generator polynomials of all MDS repeated-root constacyclic codes of arbitrary length. As applications, several new and optimal codes are provided.

Symbol-triple distance of repeated-root constacyclic codes of prime power lengths

Let [Formula: see text] be an odd prime, [Formula: see text] and [Formula: see text] be positive integers and [Formula: see text] be a nonzero element of [Formula: see text]. The [Formula: see text]-constacyclic codes of length [Formula: see text] over [Formula: see text] are linearly ordered under set theoretic inclusion as ideals of the chain ring [Formula: see text]. Using this structure, the symbol-triple distances of all such [Formula: see text]-constacyclic codes are established in this paper. All maximum distance separable symbol-triple constacyclic codes of length [Formula: see text] are also determined as an application.

Structure of some classes of repeated-root constacyclic codes of length [formula omitted

Let Fq be the finite field of order q, where q is a power of an odd prime p, and p,ℓ are distinct odd primes, and m,n,K are positive integers. In this paper, the multiplicative group Fq∗=Fq∖{0} is decomposed into mutually disjoint union of gcd(2Kℓm,q−1) cosets of the cyclic group generated by ξ2Kℓmpn, where ξ is a primitive element of Fq. With the help of this decomposition, all constacyclic codes of length 2Kℓmpn over Fq are classified into gcd(2Kℓm,q−1) disjoint classes. Accordingly, generator polynomials of constacyclic codes of length 2Kℓmpn over the finite field Fq and their duals are explicitly determined in the cases, when q≡±1(mod2K). And also some complementary-dual, self-orthogonal, self-dual, dual-containing constacyclic codes of length 2Kℓmpn over Fq are determined.

Repeated-Root Constacyclic Codes of Length $$k\ell p^s$$

In this paper, we investigate all repeated-root constacyclic codes and their duals of length $$k\ell p^s$$ over $${\mathbb {F}}_q$$ in terms of generator polynomials, where $$\ell $$ is an odd prime different from p and k is an odd prime different from both $$\ell $$ and p such that $$k=4h+1$$ for some prime h. As an application, the characterization and enumeration of all self-dual repeated-root cyclic codes of length $$2^sk\ell $$ over $${\mathbb {F}}_q$$ are obtained.

On the Structure and Distances of Repeated-Root Constacyclic Codes of Prime Power Lengths Over Finite Commutative Chain Rings

Let p be a prime, s be a positive integer, and R be a finite commutative chain ring with the characteristic as a power of p. For a unit λ ε R, λ-constacyclic codes of length ps over R are ideals of the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> -λ). In this paper, we derive necessary and sufficient conditions under which the quotient ring R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring. When R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - λ) is a chain ring, all λ-constacyclic codes of length ps over R are known. In this paper, we establish the algebraic structures of all λ-constacyclic codes of length ps over R when R[x]/(x(p) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</sup> - 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 RT weight distributions of all constacyclic codes of length ps over R. Apart from this, we derive necessary and sufficient conditions under which a constacyclic code of length ps over R is maximumdistance separable with respect to the: 1) Hamming metric; 2) symbol-pair metric; and 3) RT metric. We also provide an algorithm to decode the constacyclic codes of length ps over R using the known decoding algorithms of linear codes over finite fields with respect to the Hamming, symbol-pair, and RT metrics.

Some new constructions of isodual and LCD codes over finite fields

This paper presents some new constructions of LCD, isodual, self-dual and LCD-isodual codes based on the structure of repeated-root constacyclic codes. We first characterize repeated-root constacyclic codes in terms of their generator polynomials and lengths. Then we provide simple conditions on the existence of repeated-root codes which are either self-dual negacyclic or LCD cyclic and negacyclic. This leads to the construction of LCD, self-dual, isodual, and LCD-isodual cyclic and negacyclic codes.

Repeated-root constacyclic codes of arbitrary lengths over the Galois ring GR(p2,m)

Let [Formula: see text] be a prime, [Formula: see text] be a positive integer, and let GR[Formula: see text] be the Galois ring of characteristic [Formula: see text] and cardinality [Formula: see text]. In this paper, all repeated-root constacyclic codes of arbitrary lengths over GR[Formula: see text] their sizes and their dual codes are determined. As an application, some isodual constacyclic codes over GR[Formula: see text] are also listed. To illustrate the results, all cyclic and negacyclic codes of length 10 over [Formula: see text] are obtained.

On the Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths

Let $p$ be a prime, and $\lambda$ be a nonzero element of the finite field $\mathbb F_{p^{m}}$ . The $\lambda$ -constacyclic codes of length $p^{s}$ over $\mathbb F_{p^{m}}$ are linearly ordered under set-theoretic inclusion, i.e., they are the ideals $\langle (x-\lambda _{0})^{i} \rangle$ , $0 \leq i \leq p^{s}$ of the chain ring $[({\mathbb F_{p^{m}}[x]})/({\langle x^{p^{s}}-\lambda \rangle })]$ . This structure is used to establish the symbol-pair distances of all such $\lambda$ -constacyclic codes. Among others, all maximum distance separable symbol-pair constacyclic codes of length $p^{s}$ are obtained.

New MDS Symbol-Pair Codes From Repeated-Root Codes

Symbol-pair codes can be used to guard against pair errors over symbol-pair read channels. One of the central themes in symbol-error correction is the construction of maximal distance separable (MDS) symbol-pair codes that possess the largest possible pair-error correcting performance. In this letter, we construct three new MDS symbol-pair codes with minimum pair-distance six or seven through repeated-root constacyclic codes. In contrast with classical MDS codes, they have relatively large length.

Repeated-root constacyclic codes of length [formula omitted

In this paper, we explicitly determine the generator polynomials of all repeated-root constacyclic codes of length nlps over Fq and their dual codes, where l is an odd prime different from p, and n is an odd prime different from both l and p such that n=2h+1 for some prime h or h=1. Moreover, we give all self-dual cyclic codes of length nlps over Fq and their enumeration. All linear complementary dual (LCD) cyclic codes of length nlps over Fq and their enumeration are also obtained.

On structure and distances of some classes of repeated-root constacyclic codes over Galois rings

The structure of λ-constacyclic codes of length 2s over the Galois ring GR(2a,m) is obtained, for any unit λ of the form 4z−1, z∈GR(2a,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.

Repeated-root constacyclic codes of prime power lengths over finite chain rings

We study the algebraic structure of repeated-root λ-constacyclic codes of prime power length ps over a finite commutative chain ring R with maximal ideal 〈γ〉. It is shown that, for any unit λ of the chain ring R, there always exists an element r∈R such that λ−rps is not invertible, and furthermore, the ambient ring R[x]〈xps−λ〉 is a local ring with maximal ideal 〈x−r,γ〉. When there is a unit λ0 such that λ=λ0ps, the nilpotency index of x−λ0 in the ambient ring R[x]〈xps−λ〉 is established. When λ=λ0ps+γw, for some unit w of R, it is shown that the ambient ring R[x]〈xps−λ〉 is a chain ring with maximal ideal 〈xps−λ0〉, which in turn provides structure and sizes of all λ-constacyclic codes and their duals. Among other things, situations when a linear code over R is both α- and β-constacyclic, for different units α, β, are discussed.

Repeated-root constacyclic codes of length 3lps and their dual codes

Let p≠3 be any prime and l≠3 be any odd prime with gcd⁡(p,l)=1. The multiplicative group Fq⁎=〈ξ〉 can be decomposed into mutually disjoint union of gcd⁡(q−1,3lps) cosets over the subgroup 〈ξ3lps〉, where ξ is a primitive (q−1)th root of unity. We classify all repeated-root constacyclic codes of length 3lps over the finite field Fq into some equivalence classes by this decomposition, where q=pm, 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 3lps over Fq and their dual codes. Self-dual cyclic codes of length 3lps over Fq exist only when p=2. We give all self-dual cyclic codes of length 3⋅2sl over F2m and their enumeration. We also determine the minimum Hamming distance of these codes when gcd⁡(3,q−1)=1 and l=1.