Skew G-codes

2021 ◽  
Author(s):  
S. T. Dougherty ◽  
Serap Şahinkaya ◽  
Bahattin Yıldız
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Constacyclic Codes ◽  
Dual Codes ◽  
Frobenius Ring ◽  
Arbitrary Finite Group ◽  
Almost All ◽  
Skew Group Ring

We describe skew [Formula: see text]-codes, which are codes that are the ideals in a skew group ring, where the ring is a finite commutative Frobenius ring and [Formula: see text] is an arbitrary finite group. These codes generalize many of the well-known classes of codes such as cyclic, quasicyclic, constacyclic codes, skew cyclic, skew quasicyclic and skew constacyclic codes. Additionally, using the skew [Formula: see text]-matrices, we can generalize almost all the known constructions in the literature for self-dual codes.

scholarly journals Composite matrices from group rings, composite G-codes and constructions of self-dual codes

2021 ◽  
Author(s):  
Steven T. Dougherty ◽  
Joe Gildea ◽  
Adrian Korban ◽  
Abidin Kaya
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Group Rings ◽  
Dual Codes ◽  
Weight Enumerators ◽  
Extension Method ◽  
Frobenius Ring ◽  
Arbitrary Finite Group ◽  
Generator Matrices

AbstractIn this work, we define composite matrices which are derived from group rings. We extend the idea of G-codes to composite G-codes. We show that these codes are ideals in a group ring, where the ring is a finite commutative Frobenius ring and G is an arbitrary finite group. We prove that the dual of a composite G-code is also a composite G-code. We also define quasi-composite G-codes. Additionally, we study generator matrices, which consist of the identity matrices and the composite matrices. Together with the generator matrices, the well known extension method, the neighbour method and its generalization, we find extremal binary self-dual codes of length 68 with new weight enumerators for the rare parameters $$\gamma =7,8$$ γ = 7 , 8 and 9. In particular, we find 49 new such codes. Moreover, we show that the codes we find are inaccessible from other construction

scholarly journals On Reversible Group Rings

2006 ◽  
Vol 74 (1) ◽  
pp. 139-142 ◽  
Author(s):  
Yuanlin Li ◽  
Howard E. Bell ◽  
Colin Phipps
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Group Algebra ◽  
Associative Ring ◽  
Group Rings ◽  
Finite Rings ◽  
Arbitrary Finite Group ◽  
Algebra Over A Field

Let G be an arbitrary finite group, R be a finite associative ring with identity and RG be the group ring. We show that ℤ2Q8 is the minimal reversible group ring which is not symmetric, and we also characterise the finite rings R for which RQ8 is reversible. The first result extends a result of Gutan and Kisielewicz which shows that ℤ2Q8 is the minimal reversible group algebra over a field which is not symmetric, and it answers a question raised by Marks for the group ring case.

Homological Dimensions of Skew Group Rings

2020 ◽  
Vol 27 (02) ◽  
pp. 319-330
Author(s):  
Yueming Xiang
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Global Dimension ◽  
Group Rings ◽  
Separable Extension ◽  
Homological Dimensions ◽  
A Finite Group ◽  
Coefficient Ring ◽  
Skew Group Ring

Let R be a ring and let H be a subgroup of a finite group G. We consider the weak global dimension, cotorsion dimension and weak Gorenstein global dimension of the skew group ring RσG and its coefficient ring R. Under the assumption that RσG is a separable extension over RσH, it is shown that RσG and RσH share the same homological dimensions. Several known results are then obtained as corollaries. Moreover, we investigate the relationships between the homological dimensions of RσG and the homological dimensions of a commutative ring R, using the trivial RσG-module.

scholarly journals On the dual codes of skew constacyclic codes

2018 ◽  
Vol 12 (4) ◽  
pp. 659-679 ◽  
Author(s):  
Alexis Eduardo Almendras Valdebenito ◽  
◽  
Andrea Luigi Tironi ◽  
Keyword(s):  
Constacyclic Codes ◽  
Dual Codes

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.

scholarly journals Automorphisms of Regular Wreath Product -Groups

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Jeffrey M. Riedl
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Representation Theory ◽  
Wreath Product ◽  
Cyclic Group ◽  
Product Group ◽  
Finite Cyclic Group ◽  
Arbitrary Finite Group

We present a useful new characterization of the automorphisms of the regular wreath product group of a finite cyclic -group by a finite cyclic -group, for any prime , and we discuss an application. We also present a short new proof, based on representation theory, for determining the order of the automorphism group Aut(), where is the regular wreath product of a finite cyclic -group by an arbitrary finite -group.

Outer Partial Actions and Partial Skew Group Rings

2015 ◽  
Vol 67 (5) ◽  
pp. 1144-1160 ◽  
Author(s):  
Patrik Nystedt ◽  
Johan Öinert
Keyword(s):  
Abelian Group ◽  
Dynamical Systems ◽  
Group Ring ◽  
Group Rings ◽  
Partial Action ◽  
Unital Ring ◽  
Different Types ◽  
Outer Action ◽  
Skew Group Rings ◽  
Skew Group Ring

AbstractWe extend the classical notion of an outer action α of a group G on a unital ring A to the case when α is a partial action on ideals, all of which have local units. We show that if α is an outer partial action of an abelian group G, then its associated partial skew group ring A *α G is simple if and only if A is G-simple. This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

Algebres Simples de Groupes a Gauche

1980 ◽  
Vol 32 (1) ◽  
pp. 165-184 ◽  
Author(s):  
David Handelman
Keyword(s):  
Group Ring ◽  
Skew Group Ring

SoitRun anneau et soit Gopun sous groupe de Aut(R).Formons l'anneau de groupe à gauche (“skew group ring”)RSG,en imposant, sur le module libre à gauche avec une baseG,la multiplicationrg=grg.(On écrit Gopau lieu deGcar, siG <Inn(R) =le groupe d'automorphismes intérieurs, la conditiong–rg = rg; implique que le homomorphisme naturelest vraiment un anti-homomorphisme).Nous considérons le cas oùR=MnF, Fun corps,MnFl'anneau de matrices, d'ordren,et Gopun sous-groupe (généralement) fini, dePGL(n, F)= Inn(R).Ce travail est consacré à l'étude de la simplicité deRSG.

scholarly journals A characterization of finite p-soluble groups of p-length one by commutator identities

1981 ◽  
Vol 30 (3) ◽  
pp. 257-263 ◽  
Author(s):  
Rolf Brandl
Keyword(s):  
Finite Group ◽  
Classical Result ◽  
Soluble Groups ◽  
Similar Description ◽  
A Finite Group ◽  
Engel Condition ◽  
Almost All

AbstractA classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

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