Vol. 11, No. 2, 2018

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
The truncated and supplemented Pascal matrix and applications

Michael Hua, Steven B. Damelin, Jeffrey Sun and Mingchao Yu

Vol. 11 (2018), No. 2, 243–251
Abstract

In this paper, we introduce the k × n (with k 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.

Keywords
matroid, Pascal, network, coding, code, MDS, maximum distance separable
Mathematical Subject Classification 2010
Primary: 05B30, 05B35, 94B25
Secondary: 05B05, 05B15, 11K36, 11T71
Milestones
Received: 17 February 2016
Revised: 21 July 2016
Accepted: 15 December 2016
Published: 17 September 2017

Communicated by Jim Haglund
Authors
Michael Hua
Department of Nuclear Engineering and Radiological Sciences and Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Steven B. Damelin
Mathematical Reviews
The American Mathematical Society
Ann Arbor, MI
United States
Jeffrey Sun
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States
Mingchao Yu
College of Engineering and Computer Science
Australian National University
Canberra
Australia