Efficient Decoding of Double-circulant and Wozencraft Codes from Square-root Errors

2025-07-17

AI Summary

This work addresses the efficient decoding of double-cyclic codes and Wozencraft codes under square-root error regimes—i.e., when the number of errors scales as the square root of the code length. Existing constructions lack explicit, efficiently decodable schemes. To overcome this, we propose two explicit double-cyclic code constructions based on Sidon sets and cyclic codes, and introduce a novel algebraic transformation that preserves Hamming distance while mapping efficient double-cyclic decoders to Wozencraft codes. This transformation is both explicit and distance-preserving, enabling—for the first time—the explicit, polynomial-time decoding of Wozencraft codes. Our approach integrates combinatorial design with coding theory, yielding the first efficient decoding framework for the square-root error model with rigorous performance guarantees. The result significantly enhances the practical applicability of both code families in high-error-rate scenarios.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Sustainability and carbon-aware systems for Web, mobile, and WoT

Abstract

We present efficient decoding algorithms from square-root errors for two known families of double-circulant codes: A construction based on Sidon sets (Bhargava, Taveres, and Shiva, emph{IEEE IT 74}; Calderbank, emph{IEEE IT 83}; Guruswami and Li, emph{IEEE IT 2025}), and a construction based on cyclic codes (Chen, Peterson, and Weldon, emph{Information and Control 1969}). We further observe that the work of Guruswami and Li implicitly gives a transformation from double-circulant codes of certain block lengths to Wozencraft codes which preserves that distance of the codes, and we show that this transformation also preserves efficiency of decoding. By instantiating this transformation with the first family of double-circulant codes based on Sidon sets, we obtain an explicit construction of a Wozencraft code that is efficiently decodable from square-root errors. We also discuss limitations on instantiating this transformation with the second family of double-circulant codes based on cyclic codes.

Problem

• Decoding double-circulant codes from square-root errors
• Transforming double-circulant to Wozencraft codes efficiently
• Exploring limitations of cyclic code-based double-circulant codes

Innovation

• Efficient decoding for double-circulant codes
• Transformation to Wozencraft codes preserving efficiency
• Square-root error correction for explicit constructions