Algebraic Theoretic Properties of the Non-associative Class of (132)-Avoiding Patterns of AUNU Permutations: Applications in the Generation and Analysis of a General Cyclic Code

Chun P B,Ibrahim A.A,Garba A.I

doi:10.13189/csit.2016.040201

Abstract

The author had in [1] and based on the report as in [2], established interplay between the adjacency matrices due to Eulerian graphs constructed by the application of AUNU numbers and the generation and analysis of a general linear code. That was achieved by constructing a [5 3 2] -linear code C of size M=8. This paper reviews such a construction of a linear code as in [1] extending the approach to a larger (linear cyclic) code which is a supper code say C1 of the [5 3 2]-linear code C of size M=8, ie C⊆ C1. To achieve this, the generator matrix G as in [1] that generated C is further developed to give a matrix say G1 which now spans a larger linear code C1 of length n=5, dimension K=4 and size M=32. This is attainable by exhausting the cyclic shifts in the rows of the matrix G to give G1. It is then shown through some existing remarks and proven theorems that the linear code generated by G1 is cyclic and has generator polynomial g(x)=1+x.

Highlights

The generation of a class of linear codes by the adjacency matrix of an Eulerian graph which is due to the AUNU numbers has been reported in [1]
The special class of the (132)and(123)- avoiding class of permutation patterns was first reported by Ibrahim A.A and Audu M.S (2005) where they identified some group and graph theoretic properties of these numbers
We first make some remarks and prove some existing theorem that would be of help in the understanding of our methodology; Remark: By the generator polynomial of a linear cyclic code C, we mean a unique non-zero polynomial of minimum degree in C

Summary

Introduction

The generation of a class of linear codes by the adjacency matrix of an Eulerian graph which is due to the AUNU numbers has been reported in [1]. The special class of the (132)and(123)- avoiding class of permutation patterns was first reported by Ibrahim A.A and Audu M.S (2005) where they identified some group and graph theoretic properties of these numbers. This special class of permutation patterns has enjoyed a wide range of studies centered on applications in various areas of Mathematics in the last decade, since they were reported.

Results

Conclusion