On the dimension of ideals in group algebras, and group codes

2020 ◽  
pp. 2250024
Author(s):  
E. J. García-Claro ◽  
H. Tapia-Recillas
Keyword(s):  
Minimum Distance ◽  
Semisimple Group ◽  
Group Algebras ◽  
Mds Codes ◽  
Hamming Weight ◽  
Group Codes ◽  
Minimal Polynomials ◽  
Principal Ideals ◽  
Regular Representations ◽  
Abelian Code

Several relations and bounds for the dimension of principal ideals in group algebras are determined by analyzing minimal polynomials of regular representations. These results are used in the two last sections. First, in the context of semisimple group algebras, to compute, for any abelian code, an element with Hamming weight equal to its dimension. Finally, to get bounds on the minimum distance of certain MDS group codes. A relation between a class of group codes and MDS codes is presented. Examples illustrating the main results are provided.

scholarly journals Construction of Codes from Symmetric Groups

2019 ◽  
Vol 8 (4) ◽  
pp. 8658-8665
Keyword(s):  
Symmetric Group ◽  
Group Algebra ◽  
Minimum Distance ◽  
Symmetric Groups ◽  
Semisimple Group ◽  
Group Codes ◽  
Complete Set

Let FSn be semisimple group algebra where Sn denotes the Symmetric group of degree n. We obtain the complete set of irreducible linear idempotents of the group algebra FSn. We also find the dimension and minimum distance of the group codes over the group S

On *-clean non-commutative group rings

2016 ◽  
Vol 15 (08) ◽  
pp. 1650150 ◽  
Author(s):  
Hongdi Huang ◽  
Yuanlin Li ◽  
Gaohua Tang
Keyword(s):  
Finite Field ◽  
Local Ring ◽  
Group Algebra ◽  
Rational Numbers ◽  
Semisimple Group ◽  
Group Algebras ◽  
Group Rings ◽  
Dihedral Groups ◽  
Commutative Local Ring ◽  
Generalized Quaternion

A ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In this paper, we consider the group algebras of the dihedral groups [Formula: see text], and the generalized quaternion groups [Formula: see text] with standard involution ∗. For the non-semisimple group algebra case, we characterize the ∗-cleanness of [Formula: see text] with a prime [Formula: see text], and [Formula: see text] with [Formula: see text], where [Formula: see text] is a commutative local ring. For the semisimple group algebra case, we investigate when [Formula: see text] is ∗-clean, where [Formula: see text] is the field of rational numbers [Formula: see text] or a finite field [Formula: see text] and [Formula: see text] or [Formula: see text].

Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices

2018 ◽  
Vol 18 (3&4) ◽  
pp. 223-230
Author(s):  
Xianmang He
Keyword(s):  
Information Theory ◽  
Quantum Information ◽  
Minimum Distance ◽  
Quantum Information Theory ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Error ◽  
Generator Matrices ◽  
Quantum Mds Codes

The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

scholarly journals ON OPTIMAL QUANTUM CODES

2004 ◽  
Vol 02 (01) ◽  
pp. 55-64 ◽  
Author(s):  
MARKUS GRASSL ◽  
THOMAS BETH ◽  
MARTIN RÖTTELER
Keyword(s):  
Minimum Distance ◽  
Prime Power ◽  
Quantum Systems ◽  
Error Correcting Codes ◽  
Mds Codes ◽  
Quantum Codes ◽  
Maximum Distance Separable ◽  
Shortened Codes ◽  
Quantum Error ◽  
Quantum Mds Codes

We present families of quantum error-correcting codes which are optimal in the sense that the minimum distance is maximal. These maximum distance separable (MDS) codes are defined over q-dimensional quantum systems, where q is an arbitrary prime power. It is shown that codes with parameters 〚n, n - 2d + 2, d〛q exist for all 3≤n≤q and 1≤d≤n/2+1. We also present quantum MDS codes with parameters 〚q2, q2-2d+2, d〛q for 1≤d≤q which additionally give rise to shortened codes 〚q2-s, q2-2d+2-s, d〛q for some s.

scholarly journals Unit Groups of Semisimple Group Algebras of Abelianp-Groups over a Field

1997 ◽  
Vol 188 (2) ◽  
pp. 580-589 ◽  
Author(s):  
Nako A. Nachev ◽  
Todor Zh. Mollov
Keyword(s):  
Semisimple Group ◽  
Group Algebras

scholarly journals On Idempotents and the Number of Simple Components of Semisimple Group Algebras

2015 ◽  
Vol 19 (2) ◽  
pp. 315-333
Author(s):  
Gabriela Olteanu ◽  
Inneke Van Gelder
Keyword(s):  
Semisimple Group ◽  
Group Algebras

scholarly journals Ternary Constant Weight Codes

10.37236/1657 ◽  
2002 ◽  
Vol 9 (1) ◽  
Author(s):  
Patric R. J. Östergård ◽  
Mattias Svanström
Keyword(s):  
Lower Bounds ◽  
Minimum Distance ◽  
Upper And Lower Bounds ◽  
Hamming Weight ◽  
Maximum Cardinality ◽  
Constant Weight Codes ◽  
Constant Weight ◽  
Ternary Code

Let $A_3(n,d,w)$ denote the maximum cardinality of a ternary code with length $n$, minimum distance $d$, and constant Hamming weight $w$. Methods for proving upper and lower bounds on $A_3(n,d,w)$ are presented, and a table of exact values and bounds in the range $n \leq 10$ is given.

STRUCTURE OF SOME CLASSES OF SEMISIMPLE GROUP ALGEBRAS OVER FINITE FIELDS

2014 ◽  
Vol 51 (6) ◽  
pp. 1605-1614 ◽  
Author(s):  
Neha Makhijani ◽  
Rajendra Kumar Sharma ◽  
J.B. Srivastava
Keyword(s):  
Finite Fields ◽  
Semisimple Group ◽  
Group Algebras
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