Block-transitive algebraic geometry codes attaining the Tsfasman-Vladut-Zink bound

TL;DR

This paper introduces block-transitive algebraic geometry codes, a generalization of transitive codes, and demonstrates their ability to asymptotically attain the Tsfasman-Vladut-Zink bound over finite fields of square size.

Contribution

It establishes the existence of block-transitive codes that reach the Tsfasman-Vladut-Zink bound using towers of algebraic function fields, providing explicit parameters.

Findings

Codes attain the Tsfasman-Vladut-Zink bound over square finite fields

Explicit code lengths and lower bounds for parameters are provided

Generalizes transitive and quasi-transitive codes

Abstract

We study the asymptotic behavior of a family of algebraic geometry codes, which we call block-transitive, that generalizes the classes of transitive and quasi-transitive codes. We prove, by using towers of algebraic function fields, that there are sequences of codes in this family attaining the Tsfasman-Vladut-Zink bound over finite fields of square cardinality. We give the exact length of these codes as well as explicit lower bounds for their parameters.