Refined analysis of RGHWs of code pairs coming from Garcia-Stichtenoth's second tower

TL;DR

This paper refines the analysis of relative generalized Hamming weights (RGHWs) for algebraic geometric code pairs from Garcia-Stichtenoth's tower, improving security estimates for ramp secret sharing schemes.

Contribution

It provides refined RGHW bounds for code pairs with small codimension and improved estimates for the highest RGHW in general cases.

Findings

Refined RGHW analysis for small codimension codes

Improved upper bounds for the highest RGHW

Enhanced security estimates for secret sharing schemes

Abstract

Asymptotically good sequences of ramp secret sharing schemes were given in [Asymptotically good ramp secret sharing schemes, arXiv:1502.05507] by using one-point algebraic geometric codes defined from asymptotically good towers of function fields. Their security is given by the relative generalized Hamming weights of the corresponding codes. In this paper we demonstrate how to obtain refined information on the RGHWs when the codimension of the codes is small. For general codimension, we give an improved estimate for the highest RGHW.