Relative Generalized Hamming Weights of Cyclic Codes

TL;DR

This paper investigates the relative generalized Hamming weights of cyclic codes with two nonzeros, providing explicit formulas and examples, which are crucial for understanding security in secret sharing schemes.

Contribution

It derives two explicit formulas for RGHWs of cyclic codes with two nonzeros, expanding the theoretical understanding of their structure and security implications.

Findings

Derived two formulas for RGHWs of cyclic codes

Computed explicit examples illustrating the formulas

Enhanced understanding of security parameters in secret sharing

Abstract

Relative generalized Hamming weights (RGHWs) of a linear code respect to a linear subcode determine the security of the linear ramp secret sharing scheme based on the code. They can be used to express the information leakage of the secret when some keepers of shares are corrupted. Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems. In this paper, we investigate the RGHWs of cyclic codes with two nonzeros respect to any of its irreducible cyclic subcodes. Applying the method in the paper [arxiv.org/abs/1410.2702], we give two formulae for RGHWs of the cyclic codes. As applications of the formulae, explicit examples are computed.