Capacity-achieving codes: a review on double transitivity

TL;DR

This paper reviews the classification of codes invariant under doubly transitive groups, highlighting their capacity-achieving properties on erasure channels and discussing algebraic geometric codes.

Contribution

It provides a comprehensive review of groups and codes invariant under doubly transitive groups, advancing understanding of capacity-achieving codes.

Findings

Doubly transitive invariant codes achieve erasure channel capacity

Algebraic geometric codes form capacity-achieving families

Classification of such codes and groups is progressing

Abstract

Recently it was proved that if a linear code is invariant under the action of a doubly transitive permutation group, it achieves the capacity of erasure channel. Therefore, it is of sufficient interest to classify all codes, invariant under such permutation groups. We take a step in this direction and give a review of all suitable groups and the known results on codes invariant under these groups. It turns out that there are capacity-achieving families of algebraic geometric codes.