Abstract
In this paper, we aim at utilizing the Cayley tables demonstrated by the Authors[1] in the construction of a Generator/Parity check Matrix in standard form for a Code say C Our goal is achieved first by converting the Cayley tables in [1] using Mod2 arithmetic into a Matrix with entries from the binary field. Echelon Row operations are then performed (carried out) on the matrix in line with existing algorithms and propositions to obtain a matrix say G whose rows spans C and a matrix say H whose rows spans C⊥, the dual code of C, where G and H are given as, G = (Ik | X ) and H= ( -XT | In-k ). The report by Williem (2011) that the adjacency Matrix of a graph can be interpreted as the generator matrix of a Code [3] is in this context extended to the Cayley table which generates matrices from the permutations of points of the AUNU numbers of prime cardinality.
Highlights
In Coding theory, the generator matrix of a Code plays an important role
Once the generator matrix of a code is known, such a code can be encoded and decoded, since procedures for obtaining the parity check matrix say of the code from the generator matrix is obvious through existing algorithms and theorems
Permutation Patterns: An arrangement of the objects 1, 2, . . . , n is a sequence consisting of these objects arranged in any order
Summary
In Coding theory, the generator matrix of a Code plays an important role. Once the generator matrix of a code is known, such a code can be encoded and decoded, since procedures for obtaining the parity check matrix say of the code from the generator matrix is obvious through existing algorithms and theorems. The generation and analysis of some small classes of linear and cyclic codes from the adjacency matrices of Eulerian graphs due to AUNU patterns had been reported in [2] and [3]. This special class of the (132) and (123)-avoiding class of permutation patterns which were first reported [4], where some group and graph theoretic properties were identified, had enjoyed a wide range of applications in various areas of applied Mathematics. We review some basic concepts and propositions for the easy understanding of this paper
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