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

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 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.

scholarly journals New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Adrian Korban ◽  
Serap Şahinkaya ◽  
Deniz Ustun
Keyword(s):  
Group Rings ◽  
Type I ◽  
Identity Matrix ◽  
Dual Codes ◽  
Weight Enumerators ◽  
Composite Matrix ◽  
Binary Images ◽  
New Type ◽  
Generator Matrices

<p style='text-indent:20px;'>In this work, we define three composite matrices derived from group rings. We employ these composite matrices to create generator matrices of the form <inline-formula><tex-math id="M3">\begin{document}$ [I_n \ | \ \Omega(v)], $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M4">\begin{document}$ I_n $\end{document}</tex-math></inline-formula> is the identity matrix and <inline-formula><tex-math id="M5">\begin{document}$ \Omega(v) $\end{document}</tex-math></inline-formula> is a composite matrix and search for binary self-dual codes with parameters <inline-formula><tex-math id="M6">\begin{document}$ [36,18, 6 \ \text{or} \ 8]. $\end{document}</tex-math></inline-formula> We next lift these codes over the ring <inline-formula><tex-math id="M7">\begin{document}$ R_1 = \mathbb{F}_2+u\mathbb{F}_2 $\end{document}</tex-math></inline-formula> to obtain codes whose binary images are self-dual codes with parameters <inline-formula><tex-math id="M8">\begin{document}$ [72,36,12]. $\end{document}</tex-math></inline-formula> Many of these codes turn out to have weight enumerators with parameters that were not known in the literature before. In particular, we find <inline-formula><tex-math id="M9">\begin{document}$ 30 $\end{document}</tex-math></inline-formula> new Type I binary self-dual codes with parameters <inline-formula><tex-math id="M10">\begin{document}$ [72,36,12]. $\end{document}</tex-math></inline-formula></p>

scholarly journals Extremal binary self-dual codes of lengths 64 and 66 from four-circulant constructions over F2+uF2

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 937-945 ◽  
Author(s):  
Suat Karadeniz ◽  
Bahattin Yildiz ◽  
Nuh Aydin
Keyword(s):  
Dual Codes ◽  
Weight Enumerators ◽  
Extension Method ◽  
Extremal Codes ◽  
New Codes ◽  
First Time

A classification of all four-circulant extremal codes of length 32 over F2 + uF2 is done by using four-circulant binary self-dual codes of length 32 of minimum weights 6 and 8. As Gray images of these codes, a substantial number of extremal binary self-dual codes of length 64 are obtained. In particular a new code with ?=80 in W64,2 is found. Then applying an extension method from the literature to extremal self-dual codes of length 64, we have found many extremal binary self-dual codes of length 66. Among those, five of them are new codes in the sense that codes with these weight enumerators are constructed for the first time. These codes have the values ?=1, 30, 34, 84, 94 in W66,1.

scholarly journals Group rings which are v-HC orders and Krull orders

1991 ◽  
Vol 34 (2) ◽  
pp. 217-228 ◽  
Author(s):  
K. A. Brown ◽  
H. Marubayashi ◽  
P. F. Smith
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Prime Ideals ◽  
Group Rings

Let R be a ring and G a polycyclic-by-finite group. In this paper, it is determined, in terms of properties of R and G, when the group ring R[G] is a prime Krull order and when it is a price v-HC order. The key ingredient in obtaining both characterizations is the first author's earlier study of height one prime ideals in the ring R[G[.

scholarly journals Integral Group Rings with Nilpotent Unit Groups

1976 ◽  
Vol 28 (5) ◽  
pp. 954-960 ◽  
Author(s):  
César Polcino Milies
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Unit Element ◽  
Group Rings ◽  
Integral Group ◽  
Integral Group Rings ◽  
Group Of Units ◽  
The Subject ◽  
A Finite Group

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.The study of the nilpotency of U(RG) has been the subject of several papers.

scholarly journals On Galois projective group rings

1991 ◽  
Vol 14 (1) ◽  
pp. 149-153
Author(s):  
George Szeto ◽  
Linjun Ma
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Galois Group ◽  
Group Ring ◽  
Projective Group ◽  
Group Rings ◽  
Inner Automorphism ◽  
A Finite Group

LetAbe a ring with1,Cthe center ofAandG′an inner automorphism group ofAinduced by {Uαin​A/αin a finite groupGwhose order is invertible}. LetAG′be the fixed subring ofAunder the action ofG′.IfAis a Galcis extension ofAG′with Galois groupG′andCis the center of the subring∑αAG′UαthenA=∑αAG′Uαand the center ofAG′is alsoC. Moreover, if∑αAG′Uαis Azumaya overC, thenAis a projective group ring.

scholarly journals Units of group rings of groups of order 16

1993 ◽  
Vol 35 (3) ◽  
pp. 367-379 ◽  
Author(s):  
E. Jespers ◽  
M. M. Parmenter
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Abelian Groups ◽  
Unit Group ◽  
Integral Group Ring ◽  
Group Rings ◽  
Integral Group ◽  
Group Of Units ◽  
Abelian Case ◽  
A Finite Group

LetGbe a finite group,(ZG) the group of units of the integral group ring ZGand1(ZG) the subgroup of units of augmentation 1. In this paper, we are primarily concerned with the problem of describing constructively(ZG) for particular groupsG.This has been done for a small number of groups (see [11] for an excellent survey), and most recently Jespers and Leal [3] described(ZG) for several 2-groups. While the situation is clear for all groups of order less than 16, not all groups of order 16 were discussed in their paper. Our main aim is to complete the description of(ZG) for all groups of order 16. Since the structure of the unit group of abelian groups is very well known (see for example [10]), we are only interested in the non-abelian case.

Free Subgroups of Units in Group Rings

1984 ◽  
Vol 27 (3) ◽  
pp. 309-312 ◽  
Author(s):  
Jairo Zacarias Gonçalves
Keyword(s):  
Finite Group ◽  
Group Ring ◽  
Sufficient Conditions ◽  
Free Subgroup ◽  
Group Rings ◽  
Group Of Units ◽  
Rank 2 ◽  
A Finite Group ◽  
Necessary And Sufficient ◽  
Free Subgroups

AbstractIn this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.Some extensions of this results to infinite nilpotent and FC groups are also considered.

Principally quasi-Baer properties of group rings

2012 ◽  
Vol 49 (4) ◽  
pp. 454-465
Author(s):  
Libo Zan ◽  
Jianlong Chen
Keyword(s):  
Finite Group ◽  
Left Ideal ◽  
Group Ring ◽  
Group Rings ◽  
A Finite Group ◽  
Left Annihilator

A ring R is called left p.q.-Baer if the left annihilator of a principal left ideal is generated, as a left ideal, by an idempotent. It is first proved that for a ring R and a group G, if the group ring RG is left p.q.-Baer then so is R; if in condition G is finite then |G|−1 ∈ R. Counterexamples are given to answer the question whether the group ring RG is left p.q.-Baer if R is left p.q.-Baer and G is a finite group with |G|−1 ∈ R. Further, RD∞ is left p.q.-Baer if and only if R is left p.q.-Baer.

Sign in / Sign up

Export Citation Format

Share Document