Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex

TL;DR

This paper introduces an interleaved codex-based local decoding method for Reed-Muller codes, enabling simultaneous recovery of multiple coordinates efficiently, especially for large degrees, improving over naive repetition methods.

Contribution

The paper presents a novel interleaved codex technique for multi-point local decoding of Reed-Muller codes, reducing query complexity for recovering multiple coordinates.

Findings

Query complexity is O(q^2k) for recovering k coordinates.

Accessing k locations is cheaper than k repetitions of single-location decoding.

The method works for Reed-Muller codes with degree up to q/4.

Abstract

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is $d=\Theta({q})$, where $q$ is the code alphabet size (in fact, $d$ can be as big as $q/4$ in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously at the cost of querying $O(q^2k)$ coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing $k$ locations is in fact cheaper than repeating the procedure for accessing a single location for $k$ times. Our estimation of success error probability is based on error probability bound for $t$-wise linearly independent variables given in \cite{BR94}.