Algebraic geometry codes from higher dimensional varieties

TL;DR

This paper surveys the construction and analysis of Goppa-type algebraic geometry codes derived from higher dimensional varieties, highlighting techniques for estimating minimum distance and exploring various classes of algebraic varieties.

Contribution

It provides a comprehensive overview of existing literature on algebraic geometry codes from higher dimensional varieties and discusses connections with related coding theories.

Findings

Codes from Hermitian hypersurfaces and Grassmannians are analyzed.

Techniques for estimating minimum distance are described.

Connections with toric codes and order domains are highlighted.

Abstract

This paper is a general survey of literature on Goppa-type codes from higher dimensional algebraic varieties. The construction and several techniques for estimating the minimum distance are described first. Codes from various classes of varieties, including Hermitian hypersurfaces, Grassmannians, flag varieties, ruled surfaces over curves, and Deligne-Lusztig varieties are considered. Connections with the theories of toric codes and order domains are also briefly indicated.