The truncated and supplemented Pascal matrix and applications

Abstract

In this paper, we introduce the $k\times n$ (with $k\le n$) truncated, supplemented Pascal matrix, which has the property that any $k$ columns form a linearly independent set. This property is also present in Reed–Solomon codes; however, Reed–Solomon codes are completely dense, whereas the truncated, supplemented Pascal matrix has multiple zeros. If the maximum distance separable code conjecture is correct, then our matrix has the maximal number of columns (with the aforementioned property) that the conjecture allows. This matrix has applications in coding, network coding, and matroid theory.