On the Existence of Universally Decodable Matrices

TL;DR

This paper establishes a precise condition for the existence of universally decodable matrices, providing a constructive proof and a coding scheme that addresses an open problem in coding theory.

Contribution

It proves that the condition L ≤ q+1 is both necessary and sufficient for the existence of (L,N,q)-UDMs, resolving a recent open problem.

Findings

The condition L ≤ q+1 is necessary and sufficient for (L,N,q)-UDMs.

Provides a constructive proof and coding scheme for UDMs.

Addresses and resolves an open problem in the literature.

Abstract

Universally decodable matrices (UDMs) 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 $q$. The main result of this paper is that the simple condition $L \leq q+1$ is both necessary and sufficient for $(L,N,q)$-UDMs to exist. The existence proof is constructive and yields a coding scheme that is equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Our work resolves an open problem posed recently in the literature.