Decoding of (Interleaved) Generalized Goppa Codes
Hedongliang Liu, Sabine Pircher, Alexander Zeh, Antonia Wachter-Zeh

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 of polynomials and a Goppa polynomial . When the degree of all code locator polynomials in 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…
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Taxonomy
TopicsCoding theory and cryptography · Cryptographic Implementations and Security · Quantum-Dot Cellular Automata
