Riemann-Roch Spaces and Linear Network Codes

TL;DR

This paper constructs linear network codes using algebraic curves and Riemann-Roch spaces, focusing on curves with many points, and demonstrates their effectiveness with a new metric for decoding.

Contribution

It introduces a novel method for constructing linear network codes from algebraic curves with maximal points, enhancing decoding reliability.

Findings

Codes have minimal distance bounded from below in the metric.

Construction includes Hermitian, Suzuki, and Ree curves.

Codes are suitable for linear network coding.

Abstract

We construct linear network codes utilizing algebraic curves over finite fields and certain associated Riemann-Roch spaces and present methods to obtain their parameters. In particular we treat the Hermitian curve and the curves associated with the Suzuki and Ree groups all having the maximal number of points for curves of their respective genera. Linear network coding transmits information in terms of a basis of a vector space and the information is received as a basis of a possibly altered vector space. Ralf Koetter and Frank R. Kschischang %\cite{DBLP:journals/tit/KoetterK08} introduced a metric on the set of vector spaces and showed that a minimal distance decoder for this metric achieves correct decoding if the dimension of the intersection of the transmitted and received vector space is sufficiently large. The vector spaces in our construction have minimal distance bounded from below in the above metric making them suitable for linear network coding.