Higher genus universally decodable matrices (UDMG)
Abstract
We introduce the notion of Universally Decodable Matrices of Genus g (UDMG), which for g=0 reduces to the notion of Universally Decodable Matrices (UDM) introduced in [8]. A UDMG is a set of L matrices over a finite field, each with K rows, and a linear independence condition satisfied by collections of K+g columns formed from the initial segments of the matrices. We consider the mathematical structure of UDMGs and their relation to linear vector codes. We then give a construction of UDMG based on curves of genus g over the finite field, which is a natural generalization of the UDM constructed in [8]. We provide upper (and constructable lower) bounds for L in terms of K, q, g, and the number of columns of the matrices. We will show there is a fundamental trade off (Theorem 5.4) between L and g, akin to the Singleton bound for the minimal Hamming distance of linear vector codes.
Cite
@article{arxiv.1301.6117,
title = {Higher genus universally decodable matrices (UDMG)},
author = {Steve Limburg and David Grant and Mahesh K. Varanasi},
journal= {arXiv preprint arXiv:1301.6117},
year = {2013}
}
Comments
23 pages