scholarly journals New quantum codes from skew constacyclic codes

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ram Krishna Verma ◽  
Om Prakash ◽  
Ashutosh Singh ◽  
Habibul Islam
Keyword(s):  
Finite Field ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Chain Ring ◽  
Css Construction ◽  
Positive Integers

<p style='text-indent:20px;'>For an odd prime <inline-formula><tex-math id="M1">\begin{document}$ p $\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id="M2">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \ell $\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id="M4">\begin{document}$ \mathbb{F}_{p^m} $\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id="M5">\begin{document}$ p^{m} $\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id="M6">\begin{document}$ R_{\ell,m} = \mathbb{F}_{p^m}[v_1,v_2,\dots,v_{\ell}]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle_{1\leq i, j\leq \ell} $\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id="M7">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id="M8">\begin{document}$ p^{2^{\ell} m} $\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id="M9">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id="M10">\begin{document}$ R_{\ell,m} $\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>

Structure of repeated-root constacyclic codes of length 8ℓmpn

2019 ◽  
Vol 12 (04) ◽  
pp. 1950050
Author(s):  
Saroj Rani
Keyword(s):  
Finite Field ◽  
Linear Codes ◽  
Important Class ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Negacyclic Codes ◽  
Positive Integers

Constacyclic codes form an important class of linear codes which is remarkable generalization of cyclic and negacyclic codes. In this paper, we assume that [Formula: see text] is the finite field of order [Formula: see text] where [Formula: see text] is a power of the prime [Formula: see text] and [Formula: see text] are distinct odd primes, and [Formula: see text] are positive integers. We determine generator polynomials of all constacyclic codes of length [Formula: see text] over the finite field [Formula: see text] We also determine their dual codes.

Quantum codes over a class of finite chain ring

2016 ◽  
pp. 39-49
Author(s):  
Mustafa Sari ◽  
Irfan Siap
Keyword(s):  
Finite Field ◽  
Cyclic Codes ◽  
Gray Map ◽  
Quantum Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Quantum Error

In this study, we introduce a new Gray map which preserves the orthogonality from the chain ring F_2 [u] / (u^s ) to F^s_2 where F_2 is the finite field with two elements. We also give a condition of the existence for cyclic codes of odd length containing its dual over the ring F_2 [u] / (u^s ) . By taking advantage of this Gray map and the structure of the ring, we obtain two classes of binary quantum error correcting (QEC) codes and we finally illustrate our results by presenting some examples with good parameters.

scholarly journals Symbol-Pair Distance of Repeated-Root Constacyclic Codes of Prime Power Lengths over \({{\mathbb{F}_{p^m}[u]}/{\langle u^3\rangle}}\)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2554
Author(s):  
Mohammed E. Charkani ◽  
Hai Q. Dinh ◽  
Jamal Laaouine ◽  
Woraphon Yamaka
Keyword(s):  
Finite Field ◽  
Prime Power ◽  
Nonzero Element ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Positive Integers

Let p be a prime, s, m be positive integers, γ be a nonzero element of the finite field Fpm, and let R=Fpm[u]/⟨u3⟩ be the finite commutative chain ring. In this paper, the symbol-pair distances of all γ-constacyclic codes of length ps over R are completely determined.

scholarly journals Quantum Codes Obtained through Constacyclic Codes over Z3 + νZ3 + ωZ3 + νωZ3

2021 ◽  
Vol 14 (3) ◽  
pp. 1082-1097
Author(s):  
Jagbir Singh ◽  
Prateek Mor ◽  
Shikha . ◽  
Meena .
Keyword(s):  
Structural Properties ◽  
Gray Map ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Arbitrary Length ◽  
Negacyclic Codes ◽  
Characteristic 3

This paper is concerned with, structural properties and construction of quantum codes over Z3 by using constacyclic codes over the finite commutative non-chain ring R = Z3 + νZ3 + ωZ3 + νωZ3 where ν2 = 1, ω2 = ω, νω = νω, and Z3 is field having 3 elements with characteristic 3. A Gray map is defined between R and Z43. The parameters of quantum codes over Z3 are obtained by decomposing constacyclic codes into cyclic and negacyclic codes over Z3. As an application, some examples of quantum codes of arbitrary length, are also obtained.

scholarly journals Constacyclic and Linear Complementary Dual Codes Over Fq + uFq

2020 ◽  
Vol 70 (6) ◽  
pp. 626-632
Author(s):  
Om Prakash ◽  
Shikha Yadav ◽  
Ram Krishna Verma
Keyword(s):  
Cyclic Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Gray Image ◽  
New Class ◽  
Sharing Scheme ◽  
Lcd Codes ◽  
Reversible Cyclic Codes

This article discusses linear complementary dual (LCD) codes over ℜ = Fq+uFq(u2=1) where q is a power of an odd prime p. Authors come up with a new Gray map from ℜn to F2nq and define a new class of codes obtained as the gray image of constacyclic codes over .ℜ Further, we extend the study over Euclidean and Hermitian LCD codes and establish a relation between reversible cyclic codes and Euclidean LCD cyclic codes over ℜ. Finally, an application of LCD codes in multisecret sharing scheme is given.

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

A class of linear codes of length 2 over finite chain rings

2019 ◽  
Vol 19 (06) ◽  
pp. 2050103 ◽  
Author(s):  
Yonglin Cao ◽  
Yuan Cao ◽  
Hai Q. Dinh ◽  
Fang-Wei Fu ◽  
Jian Gao ◽  
...  
Keyword(s):  
Positive Integer ◽  
Linear Codes ◽  
Invertible Element ◽  
Irreducible Polynomial ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Finite Chain Rings ◽  
Positive Integers

Let [Formula: see text] be a finite field of cardinality [Formula: see text], where [Formula: see text] is an odd prime, [Formula: see text] be positive integers satisfying [Formula: see text], and denote [Formula: see text], where [Formula: see text] is an irreducible polynomial in [Formula: see text]. In this note, for any fixed invertible element [Formula: see text], we present all distinct linear codes [Formula: see text] over [Formula: see text] of length [Formula: see text] satisfying the condition: [Formula: see text] for all [Formula: see text]. This conclusion can be used to determine the structure of [Formula: see text]-constacyclic codes over the finite chain ring [Formula: see text] of length [Formula: see text] for any positive integer [Formula: see text] satisfying [Formula: see text].

scholarly journals Constacyclic Codes over Finite Chain Rings of Characteristic p

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 303
Author(s):  
Sami Alabiad ◽  
Yousef Alkhamees
Keyword(s):  
Local Rings ◽  
Constacyclic Codes ◽  
Finite Chain ◽  
Dual Codes ◽  
Chain Ring ◽  
Characteristic P ◽  
Arbitrary Length ◽  
Cyclotomic Cosets ◽  
Finite Chain Rings ◽  
Polynomial Representations

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.

Quantum codes from (1 − 2v)-constacyclic codes over the ring 𝔽q + u𝔽q + v𝔽q + uv𝔽q

2018 ◽  
Vol 10 (04) ◽  
pp. 1850046 ◽  
Author(s):  
Juan Li ◽  
Jian Gao ◽  
Yongkang Wang
Keyword(s):  
Structural Properties ◽  
Quantum Codes ◽  
Constacyclic Codes ◽  
Chain Ring

In this paper, structural properties of [Formula: see text]-constacyclic codes over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] is a power of some odd prime. As an application, some better quantum codes, compared with previous work, are obtained.

scholarly journals Repeated-root constacyclic codes of length 6lmpn

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Tingting Wu ◽  
Shixin Zhu ◽  
Li Liu ◽  
Lanqiang Li
Keyword(s):  
Finite Field ◽  
Disjoint Union ◽  
Multiplicative Group ◽  
Primitive Element ◽  
Constacyclic Codes ◽  
Negacyclic Codes ◽  
Positive Integers

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{F}_{q}^{*} = \mathbb{F}_{q}\setminus\{0\} $\end{document}</tex-math></inline-formula> is decomposed into a mutually disjoint union of <inline-formula><tex-math id="M4">\begin{document}$ \gcd(6l^mp^n,q-1) $\end{document}</tex-math></inline-formula> cosets over subgroup <inline-formula><tex-math id="M5">\begin{document}$ &lt;\xi^{6l^mp^n}&gt; $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M6">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is a primitive element of <inline-formula><tex-math id="M7">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>. Based on the decomposition, the structure of constacyclic codes of length <inline-formula><tex-math id="M8">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over finite field <inline-formula><tex-math id="M9">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula> and their duals is established in terms of their generator polynomials, where <inline-formula><tex-math id="M10">\begin{document}$ p\neq{3} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ l\neq{3} $\end{document}</tex-math></inline-formula> are distinct odd primes, <inline-formula><tex-math id="M12">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ n $\end{document}</tex-math></inline-formula> are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length <inline-formula><tex-math id="M14">\begin{document}$ 6l^mp^n $\end{document}</tex-math></inline-formula> over <inline-formula><tex-math id="M15">\begin{document}$ \mathbb{F}_{q} $\end{document}</tex-math></inline-formula>.</p>

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