Bounds on the minimum distance of algebraic geometry codes defined over some families of surfaces

TL;DR

This paper establishes new lower bounds on the minimum distance of algebraic geometry codes derived from certain algebraic surfaces, improving understanding of their error-correcting capabilities.

Contribution

It introduces novel bounds for algebraic geometry codes on surfaces with specific geometric properties, including those with Picard number one and embedded in projective space.

Findings

Lower bounds for codes on surfaces with nef or anti-nef canonical divisors.

Sharpened bounds for surfaces with Picard number one and no small self-intersection curves.

Specific bounds for surfaces of degree d ≥ 3 in projective 3-space.

Abstract

We prove lower bounds for the minimum distance of algebraic geometry codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Finally we specify our bounds to the case of surfaces of degree $d\geq 3$ embedded in $\mathbb{P)^3$.