ON PLOTKIN-OPTIMAL CODES OVER FINITE FROBENIUS RINGS

2006 ◽  
Vol 05 (06) ◽  
pp. 799-815 ◽  
Author(s):  
MARCUS GREFERATH ◽  
GARY McGUIRE ◽  
MICHAEL E. O'SULLIVAN
Keyword(s):  
Hadamard Matrices ◽  
Interesting Property ◽  
Optimal Codes ◽  
Frobenius Ring ◽  
Frobenius Rings ◽  
Plotkin Bound ◽  
Homogeneous Weight ◽  
Finite Frobenius Rings ◽  
Finite Frobenius Ring ◽  
The Relationship

We study the Plotkin bound for codes over a finite Frobenius ring R equipped with the homogeneous weight. We show that for codes meeting the Plotkin bound, the distribution on R induced by projection onto a coordinate has an interesting property. We present several constructions of codes meeting the Plotkin bound and of Plotkin-optimal codes. We also investigate the relationship between Butson–Hadamard matrices and codes over R meeting the Plotkin bound.

scholarly journals New bounds for codes over finite Frobenius rings

2010 ◽  
Vol 57 (2) ◽  
pp. 169-179 ◽  
Author(s):  
Eimear Byrne ◽  
Marcus Greferath ◽  
Axel Kohnert ◽  
Vitaly Skachek
Keyword(s):  
Frobenius Rings ◽  
Finite Frobenius Rings

Determining dependency and redundancy for identifying gene–gene interaction associated with complex disease

2020 ◽  
Vol 18 (05) ◽  
pp. 2050035
Author(s):  
Xiangdong Zhou ◽  
Keith C. C. Chan ◽  
Zhihua Huang ◽  
Jingbin Wang
Keyword(s):  
Genetic Variants ◽  
Conditional Independence ◽  
Complex Disease ◽  
Disease Status ◽  
Gene Interaction ◽  
Interesting Property ◽  
Gene Interactions ◽  
Definition Of ◽  
The Relationship ◽  
Novel Algorithm

As interactions among genetic variants in different genes can be an important factor for predicting complex diseases, many computational methods have been proposed to detect if a particular set of genes has interaction with a particular complex disease. However, even though many such methods have been shown to be useful, they can be made more effective if the properties of gene–gene interactions can be better understood. Towards this goal, we have attempted to uncover patterns in gene–gene interactions and the patterns reveal an interesting property that can be reflected in an inequality that describes the relationship between two genotype variables and a disease-status variable. We show, in this paper, that this inequality can be generalized to [Formula: see text] genotype variables. Based on this inequality, we establish a conditional independence and redundancy (CIR)-based definition of gene–gene interaction and the concept of an interaction group. From these new definitions, a novel measure of gene–gene interaction is then derived. We discuss the properties of these concepts and explain how they can be used in a novel algorithm to detect high-order gene–gene interactions. Experimental results using both simulated and real datasets show that the proposed method can be very promising.

scholarly journals FINITE QUASI-FROBENIUS MODULES AND LINEAR CODES

2004 ◽  
Vol 03 (03) ◽  
pp. 247-272 ◽  
Author(s):  
MARCUS GREFERATH ◽  
ALEXANDR NECHAEV ◽  
ROBERT WISBAUER
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
General Setting ◽  
Commutative Rings ◽  
Finite Ring ◽  
Weight Enumerators ◽  
Frobenius Rings ◽  
Finite Frobenius Rings

The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

Characterization of finite Frobenius rings

2001 ◽  
Vol 76 (6) ◽  
pp. 406-415 ◽  
Author(s):  
T. Honold
Keyword(s):  
Frobenius Rings ◽  
Finite Frobenius Rings

scholarly journals Application of fast Fourier transform for solving of the LPN problem over finite frobenius rings

2017 ◽  
Vol 19 (4) ◽  
Author(s):  
Антон Миколайович Олексійчук ◽  
Сергій Михайлович Ігнатенко
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Frobenius Rings ◽  
Finite Frobenius Rings ◽  
Lpn Problem

scholarly journals The homogeneous weight for Rk, related gray map and new binary quasi-cyclic codes

Filomat ◽  
2017 ◽  
Vol 31 (4) ◽  
pp. 885-897 ◽  
Author(s):  
Bahattin Yildiz ◽  
Ismail Kelebek
Keyword(s):  
Coding Theory ◽  
Cyclic Codes ◽  
Gray Map ◽  
Binary Codes ◽  
Binary Images ◽  
Frobenius Rings ◽  
Homogeneous Weights ◽  
Homogeneous Weight ◽  
Theoretical Results ◽  
Quasi Cyclic Codes

Using theoretical results about the homogeneous weights for Frobenius rings, we describe the homogeneous weight for the ring family Rk, a recently introduced family of Frobenius rings which have been used extensively in coding theory. We find an associated Gray map for the homogeneous weight using first order Reed-Muller codes and we describe some of the general properties of the images of codes over Rk under this Gray map. We then discuss quasi-twisted codes over Rk and their binary images under the homogeneous Gray map. In this way, we find many optimal binary codes which are self-orthogonal and quasi-cyclic. In particular, we find a substantial number of optimal binary codes that are quasi-cyclic of index 8, 16 and 24, nearly all of which are new additions to the database of quasi-cyclic codes kept by Chen.

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