The Truncated & Supplemented Pascal Matrix and Applications

TL;DR

This paper introduces a novel truncated and supplemented Pascal matrix with unique linear independence properties, offering potential advantages in coding, network coding, and matroid theory, and possibly achieving maximal columns under the MDS conjecture.

Contribution

The paper presents a new Pascal matrix variant with linear independence properties similar to Reed-Solomon codes but with zeros, and discusses its applications and theoretical significance.

Findings

Matrix has the property that any k columns are linearly independent.

Potential maximal number of columns if the MDS conjecture holds.

Applications in coding, network coding, and matroid theory.

Abstract

In this paper, we introduce the $k\times n$ (with $k\leq n$) 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 maximal-distance separable code conjecture is correct, then our matrix has the maximal number of columns (with the aformentioned property) that the conjecture allows. This matrix has applications in coding, network coding, and matroid theory.