On nested code pairs from the Hermitian curve

TL;DR

This paper investigates nested algebraic geometric codes from the Hermitian curve, providing improved constructions and formulas to optimize parameters for applications in secret sharing and quantum coding.

Contribution

It introduces new methods for constructing nested codes from the Hermitian curve with enhanced parameters and explicit performance estimates.

Findings

Improved code constructions for not too small codimension

Closed formula estimates on code performance

Strategies for optimizing relative minimum distances

Abstract

Nested code pairs play a crucial role in the construction of ramp secret sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum codes [Ketkar et al. 2006]. The important parameters are (1) the codimension, (2) the relative minimum distance of the codes, and (3) the relative minimum distance of the dual set of codes. Given values for two of them, one aims at finding a set of nested codes having parameters with these values and with the remaining parameter being as large as possible. In this work we study nested codes from the Hermitian curve. For not too small codimension, we present improved constructions and provide closed formula estimates on their performance. For small codimension we show how to choose pairs of one-point algebraic geometric codes in such a way that one of the relative minimum distances is larger than the corresponding non-relative minimum distance.