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

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.