## Involve: A Journal of Mathematics

- Involve
- Volume 11, Number 2 (2018), 243-251.

### The truncated and supplemented Pascal matrix and applications

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

#### 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.

#### Article information

**Source**

Involve, Volume 11, Number 2 (2018), 243-251.

**Dates**

Received: 17 February 2016

Revised: 21 July 2016

Accepted: 15 December 2016

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.involve/1513775060

**Digital Object Identifier**

doi:10.2140/involve.2018.11.243

**Mathematical Reviews number (MathSciNet)**

MR3733955

**Zentralblatt MATH identifier**

06817018

**Subjects**

Primary: 05B30: Other designs, configurations [See also 51E30] 05B35: Matroids, geometric lattices [See also 52B40, 90C27] 94B25: Combinatorial codes

Secondary: 05B05: Block designs [See also 51E05, 62K10] 05B15: Orthogonal arrays, Latin squares, Room squares 11K36: Well-distributed sequences and other variations 11T71: Algebraic coding theory; cryptography

**Keywords**

matroid Pascal network coding code MDS maximum distance separable

#### Citation

Hua, Michael; Damelin, Steven B.; Sun, Jeffrey; Yu, Mingchao. The truncated and supplemented Pascal matrix and applications. Involve 11 (2018), no. 2, 243--251. doi:10.2140/involve.2018.11.243. https://projecteuclid.org/euclid.involve/1513775060