scholarly journals Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉

2018 ◽  
Vol 16 (1) ◽  
pp. 490-497
Author(s):  
Xiying Zheng ◽  
Bo Kong
Keyword(s):  
Linear Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Macwilliams Identities

AbstractIn this paper, we study linear codes over ring Rk = 𝔽pm[u1, u2,⋯,uk]/〈$\begin{array}{} u^{2}_{i} \end{array} $ = ui, uiuj = ujui〉 where k ≥ 1 and 1 ≤ i, j ≤ k. We define a Gray map from $\begin{array}{} R_{k}^n\,\,\text{to}\,\,{\mathbb F}_{p^m}^{2^kn} \end{array} $ and give the generator polynomials of constacyclic codes over Rk. We also study the MacWilliams identities of linear codes over Rk.

scholarly journals Linear, cyclic and constacyclic codes over S4 = F2 + uF2 + u2F2 + u3F2

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 897-906
Author(s):  
Ödemiş Özger ◽  
Ümmü Kara ◽  
Bahattin Yıldız
Keyword(s):  
Linear Codes ◽  
Gray Map ◽  
Binary Codes ◽  
Constacyclic Codes ◽  
Weight Enumerators ◽  
Lee Weight ◽  
Macwilliams Identities

In this work, linear codes over the ring S4 = F2 + uF2 + u2F2 + u3F2 are considered. The Lee weight and gray map for codes over S4 are defined and MacWilliams identities for the complete, the symmetrized and the Lee weight enumerators are obtained. Cyclic and (1 + u2)-constacyclic codes over S4 are studied, as a result of which a substantial number of optimal binary codes of different lengths are obtained as the Gray images of cyclic and constacyclic codes over S4.

Gray images of (1 − u)-constacyclic codes over the ring 𝔽pk + u𝔽pk + v𝔽pk + uv𝔽pk

2018 ◽  
Vol 11 (02) ◽  
pp. 1850026
Author(s):  
Pramod Kumar Kewat ◽  
Sarika Kushwaha
Keyword(s):  
Positive Integer ◽  
Linear Code ◽  
Linear Codes ◽  
Gray Map ◽  
Constacyclic Codes ◽  
Arbitrary Length ◽  
Optimal Linear ◽  
Optimal Linear Codes

Let [Formula: see text], where [Formula: see text], [Formula: see text], [Formula: see text] is a prime and [Formula: see text] is a positive integer. We define a gray map from a linear code of length [Formula: see text] over the ring [Formula: see text] to a linear code of length [Formula: see text] over the field [Formula: see text]. In this paper, we characterize the gray images of [Formula: see text]-constacyclic codes of an arbitrary length over the ring [Formula: see text] in terms of quasicyclic codes over [Formula: see text]. We obtain some optimal linear codes over [Formula: see text] as gray images.

On the linear codes over the ring Rp

2016 ◽  
Vol 08 (02) ◽  
pp. 1650036 ◽  
Author(s):  
Abdullah Dertli ◽  
Yasemin Cengellenmis ◽  
Senol Eren
Keyword(s):  
Prime Number ◽  
Linear Codes ◽  
Cyclic Codes ◽  
Error Correcting Codes ◽  
Constacyclic Codes ◽  
Nontrivial Automorphism ◽  
Macwilliams Identities ◽  
Skew Cyclic Codes ◽  
Quantum Error ◽  
Quasi Cyclic Codes

Some results are generalized on linear codes over [Formula: see text] in [15] to the ring [Formula: see text], where [Formula: see text] is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over [Formula: see text] are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over [Formula: see text]. A nontrivial automorphism [Formula: see text] on the ring [Formula: see text] is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over [Formula: see text] are introduced. The number of distinct skew cyclic codes over [Formula: see text] is given. The Gray images of skew codes over [Formula: see text] are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over [Formula: see text] are introduced. MacWilliams identities of linear codes over [Formula: see text] are given.

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>

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.

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.

THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES

2015 ◽  
Vol 76 (1) ◽  
Author(s):  
Bao Xiaomin
Keyword(s):  
Linear Codes ◽  
Macwilliams Identity ◽  
Macwilliams Identities

We use derivatives to prove the equivalences between MacWilliams identity and its four equivalent forms, and present new interpretations for the four equivalent forms.

Some results on quadratic residue codes over the ring Fp + vFp + v2Fp + v3Fp

2017 ◽  
Vol 09 (03) ◽  
pp. 1750035
Author(s):  
Jian Gao ◽  
Fanghui Ma
Keyword(s):  
Linear Codes ◽  
Quadratic Residue ◽  
The Self ◽  
Dual Code ◽  
Gray Map ◽  
Chain Ring ◽  
Qr Codes ◽  
Quadratic Residue Codes ◽  
Self Duality ◽  
Gray Maps

Quadratic residue (QR) codes and their extensions over the finite non-chain ring [Formula: see text] are studied, where [Formula: see text], [Formula: see text] is an odd prime and [Formula: see text]. A class of Gray maps preserving the self-duality of linear codes from [Formula: see text] to [Formula: see text] is given. Under a special Gray map, a self-dual code [Formula: see text] over [Formula: see text], a formally self-dual code [Formula: see text] over [Formula: see text] and a formally self-dual code [Formula: see text] over [Formula: see text] are obtained from extended QR codes.

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