Decoding of (Interleaved) Generalized Goppa Codes

TL;DR

This paper investigates binary generalized Goppa codes, deriving parity-check matrices, improving minimum distance bounds, and presenting decoding algorithms for both standard and interleaved codes, with implications for post-quantum cryptography.

Contribution

It introduces a new parity-check matrix, improved minimum distance bounds, and decoding algorithms for generalized Goppa codes, including interleaved variants, relevant to cryptography.

Findings

Derived a parity-check matrix for generalized Goppa codes.

Established a lower bound on the minimum Hamming distance.

Developed a quadratic-time decoding algorithm for errors up to half the minimum distance.

Abstract

Generalized Goppa codes are defined by a code locator set $\mathcal{L}$ of polynomials and a Goppa polynomial $G(x)$. When the degree of all code locator polynomials in $\mathcal{L}$ is one, generalized Goppa codes are classical Goppa codes. In this work, binary generalized Goppa codes are investigated. First, a parity-check matrix for these codes with code locators of any degree is derived. A careful selection of the code locators leads to a lower bound on the minimum Hamming distance of generalized Goppa codes which improves upon previously known bounds. A quadratic-time decoding algorithm is presented which can decode errors up to half of the minimum distance. Interleaved generalized Goppa codes are introduced and a joint decoding algorithm is presented which can decode errors beyond half the minimum distance with high probability. Finally, some code parameters and how they apply to the Classic McEliece post-quantum cryptosystem are shown.