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

TL;DR

This paper introduces efficient decoding algorithms for specific double-circulant codes from square-root errors, transforming certain codes into Wozencraft codes while preserving decoding efficiency and code distance.

Contribution

It provides a novel transformation from double-circulant codes to Wozencraft codes that maintains decoding efficiency and code distance, with explicit constructions based on Sidon sets.

Findings

Efficient decoding algorithms for double-circulant codes from square-root errors.

Transformation preserves code distance and decoding efficiency.

Explicit construction of a Wozencraft code decodable from square-root errors.

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.