A generalization of the pascal matrix and an application to coding theory

Abstract

Suppose that $m,k,t$ are integers with $m,k\ge 1$ and $0\le t$ and A is the $k\times k$ matrix with the $(i,j)$-entry $(\genfrac{}{}{0ex}{}{t+(j-1)m}{i-1})$. When t = 0 and m = 1, this is the upper triangular Pascal matrix. Here, first we study the properties of this matrix, in particular, we find its determinant and its LDU decomposition and also study its inverse. Then by using this matrix we present a generalization of the Mattson-Solomon transform and a polynomial formulation for its inverse to the case that the sequence length and the characteristic of the base field are not coprime. At the end, we use this generalized Mattson-Solomon transform to present a lower bound on the length of repeated root cyclic codes, which can be seen as a generalization of the BCH bound.

KEYWORDS:

• Upper triangular pascal matrix
• mattson-solomon polynomial
• generalized discrete fourier transform
• hasse derivative
• repeated-root cyclic codes

2020 MATHEMATICAL SUBJECT CLASSIFICATIONS:

• 15B36
• 15A15
• 05A10
• 15B33
• 11T71

Acknowledgments

The authors would like to thank the reviewers for their nice comments which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).