An Explicit Construction of Universally Decodable Matrices

TL;DR

This paper presents an explicit construction of universally decodable matrices using Pascal's triangle, applicable for any non-zero integers L and n with prime power q, connecting to Reed-Solomon and Reed-Muller codes.

Contribution

It provides the largest explicit construction of universally decodable matrices for parameters L and n, using properties of Hasse derivatives and linking to well-known coding theories.

Findings

Construction is valid for L ≤ q+1, the maximum possible range.

Uses properties of Hasse derivatives for proof.

Connects universally decodable matrices to Reed-Solomon and Reed-Muller codes.

Abstract

Universally decodable matrices can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers $L$ and $n$ and a prime power $q$. Based on Pascal's triangle we give an explicit construction of universally decodable matrices for any non-zero integers $L$ and $n$ and any prime power $q$ where $L \leq q+1$. This is the largest set of possible parameter values since for any list of universally decodable matrices the value $L$ is upper bounded by $q+1$, except for the trivial case $n = 1$. For the proof of our construction we use properties of Hasse derivatives, and it turns out that our construction has connections to Reed-Solomon codes, Reed-Muller codes, and so-called repeated-root cyclic codes. Additionally, we show how universally decodable matrices can be modified so that they remain universally decodable matrices.