Network-Error Correcting Codes using Small Fields

TL;DR

This paper presents an efficient algorithm for constructing network-error correcting codes over small finite fields, reducing computational complexity and improving practicality for large networks.

Contribution

It introduces a novel algorithm that transforms existing network-error correcting codes into codes over smaller fields, enhancing efficiency and broadening applicability.

Findings

The algorithm reduces field size requirements for network-error correcting codes.

It speeds up the process of finding suitable irreducible polynomials using algebraic properties.

The method generalizes and improves upon previous algorithms by Ebrahimi and Fragouli.

Abstract

Existing construction algorithms of block network-error correcting codes require a rather large field size, which grows with the size of the network and the number of sinks, and thereby can be prohibitive in large networks. In this work, we give an algorithm which, starting from a given network-error correcting code, can obtain another network code using a small field, with the same error correcting capability as the original code. An algorithm for designing network codes using small field sizes proposed recently by Ebrahimi and Fragouli can be seen as a special case of our algorithm. The major step in our algorithm is to find a least degree irreducible polynomial which is coprime to another large degree polynomial. We utilize the algebraic properties of finite fields to implement this step so that it becomes much faster than the brute-force method. As a result the algorithm given by Ebrahimi and Fragouli is also quickened.