Decoding of Lifted Affine-Invariant Codes

TL;DR

This paper introduces a simple decoding method for lifted affine-invariant codes, enabling near-half minimum distance decoding and high error correction capability for long q-ary codes, enhancing their practical utility.

Contribution

It presents a straightforward bounded distance decoder for lifted affine-invariant codes and demonstrates their ability to correct almost all high-weight error patterns.

Findings

Decoder guarantees correction up to nearly half the minimum distance.

Long q-ary codes can correct almost all errors of relative weight close to 1.

Lifted affine-invariant codes maintain high rate and locality properties.

Abstract

Lifted Reed-Solomon codes, a subclass of lifted affine-invariant codes, have been shown to be of high rate while preserving locality properties similar to generalized Reed-Muller codes, which they contain as subcodes. This work introduces a simple bounded distance decoder for (subcodes of) lifted affine-invariant codes that is guaranteed to decode up to almost half of their minimum distance. Further, long $q$-ary lifted affine-invariant codes are shown to correct almost all error patterns of relative weight $\frac{q-1}{q}-\epsilon$ for $\epsilon>0$.