Λ-constacyclic Codes Research Articles

A class of constacyclic codes of length 2pˢ over Fp^m[u,v] / ⟨u², v², uv − vu⟩

Let p be an odd prime number. This paper investigates λ-constacyclic codes of length 2ps over the ring Rᵤ,ᵥ = Fpm[u,v] / ⟨u², v², uv − vu⟩, where λ = ψ + ρu + ηv + τuv, with ψ, ρ, η, τ ∈ Fpm, ψ ∈ F*pm, and ρ, η not both zero. When λ = μ² for some μ ∈ Rᵤ,ᵥ, the structures of all λ-constacyclic codes of length 2ps over Rᵤ,ᵥ are determined and can be expressed as the sum of a (−μ)-constacyclic code and a μ-constacyclic code of length ps. When λ is not a square, we classify these codes into four distinct types, provide the number of codewords for each type, and give the details of their dual codes.

A tight upper bound on the number of non-zero weights of a constacyclic code

For a simple-root λ-constacyclic code C over Fq, let 〈ρ〉 and 〈ρ,M〉 be the subgroups of the automorphism group of C generated by the cyclic shift ρ, and by the cyclic shift ρ and the scalar multiplication group M, respectively. Let NG(C⁎) be the number of orbits of a subgroup G of the automorphism group of C acting on C⁎=C﹨{0}. In this paper, we establish explicit formulas for N〈ρ〉(C⁎) and N〈ρ,M〉(C⁎). Consequently, we derive a upper bound on the number of non-zero weights of C. We present some irreducible and reducible λ-constacyclic codes, which show that the upper bound is tight. A sufficient condition to guarantee N〈ρ〉(C⁎)=N〈ρ,M〉(C⁎) is presented.

On Hamming distance distributions of repeated-root constacyclic codes of length 3ps over [formula omitted

Let p≠3 be any prime. The algebraic structures of all λ-constacyclic codes of length 3ps over the finite commutative chain ring R=Fpm+uFpm are provided by classifying all ideals of the local ring R[x]〈x3ps−λ〉. Using these structures, in this paper, the exact values of Hamming distances of all such λ-constacyclic codes of length 3ps are established. As an application, we identify all maximal distance separable λ-constacyclic codes of length 3ps.

Quantum codes from constacyclic codes over a semi-local ring

Let R r = F q + v 1 F F q + ··· + v r F q , where q is the power of prime, v i 2 = v i , v i v j = v j v i = 0 for 1 ≤ i, j ≤ r and r ≥ 1. In this paper, the structure of λ-constacyclic codes over the ring R r is studied and a Gray map ϕ from R r n to F q ( r +1) n is given. The necessary and sufficient conditions for these codes to contain their Euclidean duals are determined. As an application, we obtain many new better quantum codes from dual-containing λ-constacyclic codes over R r , for r = 1, 2, 3, that improve the known existing quantum codes.

A note on constacyclic codes over the ring z_3[u, v

In this paper, we study λ-constacyclic codes over the ring R = Z 3 [u, v]/ for λ = (1 + u),(2 + 2u) and 2. We introduce a Gray map from R to Z 3 3 and show that the Gray image of a cyclic code is a quasi-cyclic code of index 3. It is proved that the Gray image of λ-constacyclic code over R is permutation equivalent to either quasi-cyclic or quasi-twisted code according to the value of λ. Moreover, we determine the structure of (1+u)-constacyclic codes for an odd length n over R and give some suitable examples.

Constacyclic Codes over Finite Chain Rings of Characteristic p

Let R be a finite commutative chain ring of characteristic p with invariants p,r, and k. In this paper, we study λ-constacyclic codes of an arbitrary length N over R, where λ is a unit of R. We first reduce this to investigate constacyclic codes of length ps (N=n1ps,p∤n1) over a certain finite chain ring CR(uk,rb) of characteristic p, which is an extension of R. Then we use discrete Fourier transform (DFT) to construct an isomorphism γ between R[x]/<xN−λ> and a direct sum ⊕b∈IS(rb) of certain local rings, where I is the complete set of representatives of p-cyclotomic cosets modulo n1. By this isomorphism, all codes over R and their dual codes are obtained from the ideals of S(rb). In addition, we determine explicitly the inverse of γ so that the unique polynomial representations of λ-constacyclic codes may be calculated. Finally, for k=2 the exact number of such codes is provided.

Non-invertible-element constacyclic codes over finite PIRs

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ of R. We study the non-invertible-element constacyclic codes (NIE-constacyclic codes) over finite principal ideal rings (PIRs). We determine the algebraic structures of all NIE-constacyclic codes over finite chain rings, give the unique form of the sets of the defining polynomials and obtain their minimum Hamming distances. A general form of the duals of NIE-constacyclic codes over finite chain rings is also provided. In particular, we give a necessary and sufficient condition for the dual of an NIE-constacyclic code to be an NIE-constacyclic code. Using the Chinese Remainder Theorem, we study the NIE-constacyclic codes over finite PIRs. Furthermore, we construct some optimal NIE-constacyclic codes over finite PIRs in the sense that they achieve the maximum possible minimum Hamming distances for some given lengths and cardinalities.

Constructing MDS Galois self-dual constacyclic codes over finite fields

As a further development and deepening of two examples in the paper (Fan and Zhang, 2017), we construct MDS Galois self-dual constacyclic codes over finite fields. First, we give the definition of normal MDS constacyclic codes and explore their existence conditions. Further, we give existence conditions for normal MDS Galois self-dual constacyclic codes. Based on the latter conditions, we classify and construct all normal MDS Galois self-dual λ-constacyclic codes over Fq of length n, where q=pe is an odd prime power, gcd(q,n)=1 and λ∈Fq∗.

A note on constacyclic codes over the ring z_3[u, v]/<u^2 −u, v^2, uv, vu>

In this paper, we study λ-constacyclic codes over the ring R = Z3[u, v]/<u2 − u, v2, uv, vu> for λ = (1 + u),(2 + 2u) and 2. We introduce a Gray map from R to Z33 and show that the Gray image of a cyclic code is a quasi-cyclic code of index 3. It is proved that the Gray image of λ-constacyclic code over R is permutation equivalent to either quasi-cyclic or quasi-twisted code according to the value of λ. Moreover, we determine the structure of (1+u)-constacyclic codes for an odd length n over R and give some suitable examples.

Hamming distances of constacyclic codes of length 3ps and optimal codes with respect to the Griesmer and Singleton bounds

Let p≠3 be a prime, s, m be positive integers, and λ be a nonzero element of the finite field Fpm. In [22] and [20], when the generator polynomials have one or two different irreducible factors, the Hamming distances of λ-constacyclic codes of length 3ps over Fpm have been considered. In this paper, we obtain that the Hamming distances of the repeated-root λ-constacyclic codes of length lps can be determined by the Hamming distances of the simple-root γ-constacyclic codes of length l, where l is a positive integer and λ=γps. Based on this result, the Hamming distances of the repeated-root λ-constacyclic codes of length 3ps are given when the generator polynomials have three different irreducible factors. Hence, the Hamming distances of all such constacyclic codes are determined. As an application, we obtain all optimal λ-constacyclic codes of length 3ps with respect to the Griesmer bound and the Singleton bound. Among others, several examples show that some of our codes have the best known parameters with respect to the code tables in [19].

Notes on LCD Codes Over Frobenius Rings

Let R be a commutative Frobenius local ring. A result that the injective hull of an LCD code C over R. is free of dimension 1(C), where 1(C) is the minimum over the cardinalities of the generating sets of C, is proved in this correspondence. Applying this result, a concise proof for the main result in a recent paper by Sanjit Bhowmick et al. is derived. Furthermore, the LCD λ-constacyclic codes with λ being a unit, π(λ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) = 1 and λ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ≠ 1, where π is the natural projection of R. to its residue field, are characterized, as another application of our result.

Repeated-root constacyclic codes of length 2ps over $\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}$

Let λ be a unit of the finite commutative chain ring $R=\mathbb {F}_{p^{m}}+u\mathbb {F}_{p^{m}}+u^{2}\mathbb {F}_{p^{m}}=\{\alpha +u\upbeta +u^{2}\gamma : \alpha ,\upbeta ,\gamma \in \mathbb { F}_{p^{m}}\}$ with u3 = 0, where p is an odd prime and m is a positive integer. In this paper, we consider any λ-constacyclic codes of length 2ps over R. In the case of square λ = δ2, where δ ∈ R, the algebraic structures of all λ-constacyclic codes of length 2ps over R are determined by the Chinese Remainder Theorem in terms of polynomial generators. Precisely, each λ-constacyclic code of length 2ps is represented as a direct sum of a (−δ)-constacyclic code and a δ-constacyclic code of length ps. In the case of non-square λ = α + uβ + u2γ or λ = α + uβ, where $\alpha ,\upbeta ,\gamma \in \mathbb {F}_{p^{m}}\setminus \{0\}$ , it is shown that the ring $\mathcal {R}=\frac {R[x]}{\langle x^{2p^{s}}-\lambda \rangle }$ is a chain ring. In the case of non-square $\lambda =\alpha \in \mathbb {F}_{p^{m}}\setminus \{0\}$ , it turns out that λ-constacyclic codes are classified into 8 distinct types of ideals, and the detailed structures of ideals in each type are provided.

Z 4 Z 4 [u]-ADDITIVE CYCLIC AND CONSTACYCLIC CODES

We study mixed alphabet cyclic and constacyclic codes over the two alphabets Z 4 , the ring of integers modulo 4, and its quadratic extension Z 4 [u] = Z 4 + uZ 4 , u 2 = 0. Their generator polynomials and minimal spanning sets are obtained. Further, under new Gray maps, we nd cyclic, quasi-cyclic codes over Z 4 as the Gray images of both λ-constacyclic and skew λ-constacyclic codes over Z 4 [u]. Moreover, it is proved that the Gray images of Z 4 Z 4 [u]-additive constacyclic and skew Z 4 Z 4 [u]-additive constacyclic codes are generalized quasi-cyclic codes over Z 4. Finally, several new quaternary linear codes are obtained from these cyclic and constacyclic codes.

Some Results on Images of a Class of λ-Constacyclic Codes Over Finite Fields

In this paper, we show that any λ-constacyclic code over F qm is a λ-constacyclic code over F q under some special maps. Moreover, we show that the images of these λ_constacyclic codes can be put into the concatenated form.

On the [formula omitted]-distance of repeated-root constacyclic codes of prime power lengths

Let p be a prime, s, m be positive integers, λ be a nonzero element of the finite field Fpm. The b-distance generalizes the Hamming distance (b=1), and the symbol-pair distance (b=2). While the Hamming and symbol-pair distances of all λ-constacyclic codes of length ps are completely determined, the general b-distance of such codes was left opened. In this paper, we provide a new technique to establish the b-distance of all λ-constacyclic codes of length ps, where 1≤b≤⌊p2⌋. As an application, all MDS b-symbol constacyclic codes of length ps over Fpm are obtained.

On the symbol-pair distances of repeated-root constacyclic codes of length [formula omitted

Let p be an odd prime, and λ be a nonzero element of the finite field Fpm. The λ-constacyclic codes of length 2ps over Fpm are classified as the ideals of quotient ring Fpm[x]〈x2ps−λ〉 in terms of their generator polynomials. Based on these generator polynomials, the symbol-pair distances of all such λ-constacyclic codes of length 2ps are obtained in this paper. As an application, all MDS symbol-pair constacyclic codes of length 2ps over Fpm are established, which produce many new MDS symbol-pair codes with good parameters.

On a Class of Constacyclic Codes of Length 4ps over Fpm[u]〈 ua 〉

For any odd prime p such that pm ≡ 3 (mod 4), consider all units Λ of the finite commutative chain ring [Formula: see text] that have the form Λ = Λ0 + uΛ1 + ⋯ + ua−1 Λa−1, where Λ0, Λ1, …, Λa−1 ∊ 𝔽pm, Λ0 ≠ 0, Λ1 ≠ 0. The class of Λ-constacyclic codes of length 4ps over ℛa is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a −λ-constacyclic code and a λ-constacyclic code of length 2ps. In the main case that the unit Λ is not a square, we prove that the polynomial x4 − λ0 can be decomposed as a product of two quadratic irreducible and monic coprime factors, where [Formula: see text]. From this, the ambient ring [Formula: see text] is proven to be a principal ideal ring, whose maximal ideals are ⟨x2 + 2ηx + 2η2⟩ and ⟨x2 − 2ηx + 2η2⟩, where λ0 = −4η4. We also give the unique self-dual Type 1 Λ-constacyclic codes of length 4ps over ℛa. Furthermore, conditions for a Type 1 Λ-constacyclic code to be self-orthogonal and dual-containing are provided.

RT distance and weight distributions of Type 1 constacyclic codes of length 4ps over Fpm[u

© Tubitak. For any odd prime p such that p m ≡ 1 (mod 4), the class of Λ-constacyclic codes of length 4p s over the finite commutative chain ring R a =F p m[u]/〈u a 〉 =F p m + uF p m + . . . + ua -1 F p m, for all units Λ of R a that have the form Λ = Λ 0 +uΛ 1 +. . .+u a-1 Λ a-1 , where Λ 0 ;Λ 1 ;: :: ;Λ a-1 ∈ F p m, Λ 0 ≠0; Λ 1 ≠0, is investigated. If the unit Λ is a square, each Λ-constacyclic code of length 4ps is expressed as a direct sum of a -λ-constacyclic code and a λ-constacyclic code of length 2p s . In the main case that the unit Λ is not a square, we show that any nonzero polynomial of degree < 4 over F p m is invertible in the ambient ring R a [x] /〈x 4ps -Λ〉 and use it to prove that the ambient ring 〈 a [x] /x 4ps -Λ〉 is a chain ring with maximal ideal 〈x 4 - λ 0 〉, where λ 0ps = λ 0 : As an application, the number of codewords and the dual of each λ-constacyclic code are provided. Furthermore, we get the Rosenbloom-Tsfasman (RT) distance and weight distributions of such codes. Using these results, the unique MDS code with respect to the RT distance is identified.

On the Hamming distances of repeated-root constacyclic codes of length [formula omitted

Let p be an odd prime, s, m be positive integers, γ,λ be nonzero elements of the finite field Fpm such that γps=λ. In this paper, we show that, for any positive integer η, the Hamming distances of all repeated-root λ-constacyclic codes of length ηps can be determined by those of certain simple-root γ-constacyclic codes of length η. Using this result, Hamming distances of all constacyclic codes of length 4ps are obtained. As an application, we identify all MDS λ-constacyclic codes of length 4ps.

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.