Higher genus universally decodable matrices (UDMG)

TL;DR

This paper introduces Universally Decodable Matrices of Genus g (UDMG), generalizing UDMs by incorporating algebraic curves of genus g, and explores their structure, bounds, and relation to linear codes.

Contribution

It defines UDMG, relates it to algebraic curves, and establishes bounds and trade-offs, extending the concept of UDMs to higher genus cases.

Findings

UDMG generalizes UDM with algebraic curves of genus g.

Bounds for the number of matrices L in UDMG are established.

A fundamental trade-off between L and g is proven, similar to the Singleton bound.

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.