Codes and the Cartier Operator

TL;DR

This paper introduces Cartier Codes, a new class of algebraic-geometry-based codes constructed via the Cartier operator, extending classical Goppa codes and improving bounds on code parameters.

Contribution

It presents a novel construction of codes from algebraic curves using the Cartier operator, generalizing Goppa codes and linking to subfield subcodes of algebraic geometry codes.

Findings

Cartier codes are contained within subfield subcodes of algebraic geometry codes.

The construction extends properties of classical Goppa codes.

New bounds on dimension and minimum distance are established.

Abstract

In this article, we present a new construction of codes from algebraic curves. Given a curve over a non-prime finite field, the obtained codes are defined over a subfield. We call them Cartier Codes since their construction involves the Cartier operator. This new class of codes can be regarded as a natural geometric generalisation of classical Goppa codes. In particular, we prove that a well-known property satisfied by classical Goppa codes extends naturally to Cartier codes. We prove general lower bounds for the dimension and the minimum distance of these codes and compare our construction with a classical one: the subfield subcodes of Algebraic Geometry codes. We prove that every Cartier code is contained in a subfield subcode of an Algebraic Geometry code and that the two constructions have similar asymptotic performances. We also show that some known results on subfield subcodes of Algebraic Geometry codes can be proved nicely by using properties of the Cartier operator and that some known bounds on the dimension of subfield subcodes of Algebraic Geometry codes can be improved thanks to Cartier codes and the Cartier operator.