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In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithms. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and the encoding of a message polynomial is the collection of residues of that polynomial modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis. Our framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Univariate Multiplicity codes as well as the less-studied Additive Folded Reed-Solomon codes, and lead to a large family of codes that were not previously known/studied. More significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal-theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that a good bound on this distance leads to a capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. In the specific case of affine Folded Reed-Solomon codes, our framework shows that they are efficiently list-decodable up to capacity (for appropriate setting of the parameters), thereby unifying the previous results for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.
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This content will become publicly available on January 1, 2027
Unique Decoding of Reed–Solomon and Related Codes for Semi-Adversarial Errors
Abstract: Motivated by recent developments in coding theory, particular in list-decoding, we introduce a new error model which we call semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. For interleaved Reed-Solomon (IRS), folded Reed-Solomon (FRS) and univariate multiplicity codes, we design decoding algorithms running in near-linear time for most mixtures of random and adversarial errors. Our analysis matches the information-theoretic optimum for semi-adversarial errors. Our algorithm for interleaved Reed-Solomon codes is an improved implementation of the decoding algorithm by Bleichenbacher-Kiayias-Yung (BKY) for fully random errors. We use a novel monomial-tracking technique to analyze its performance in this new semi-adversarial errors. Inspired by the BKY algorithm, we use novel interpolations to extend our approach to the settings of folded Reed-Solomon and multiplicity codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. Our new decoders for FRS and multiplicity codes replace the sophisticated root-finding step in traditional algorithms, such as the Guruswami-Wang algorithm, with a straightforward polynomial long division. Analysis of these algorithms requires more robust monomial-tracking arguments than IRS codes.
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- Award ID(s):
- 2236931
- PAR ID:
- 10708720
- Publisher / Repository:
- Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- Date Published:
- Volume:
- 374
- ISSN:
- 1868-8969
- ISBN:
- 978-3-95977-428-4
- Page Range / eLocation ID:
- 43:1-43:19
- Subject(s) / Keyword(s):
- coding theory interleaved codes Reed-Solomon codes semi-random models unique decoding Mathematics of computing → Coding theory Computing methodologies → Algebraic algorithms
- Format(s):
- Medium: X Size: 19 pages; 967254 bytes Other: application/pdf
- Size(s):
- 19 pages 967254 bytes
- Right(s):
- Creative Commons Attribution 4.0 International license; info:eu-repo/semantics/openAccess
- Sponsoring Org:
- National Science Foundation
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