An Explicit Construction of Universally Decodable Matrices

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 ≤ 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. Index terms — Universally decodable matrices, coding for slow fading channels, Pascal’s triangle, rank condition, linear independence, Reed-Solomon codes, Reed-Muller codes, repeated-root cyclic codes, Hasse derivative.