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arXiv:1702.02671

On the Local Correctabilities of Projective Reed-Muller Codes

Sian-Jheng Lin

Published Feb 9, 2017• 0 citations
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Abstract

In this paper, we show that the projective Reed-Muller~(PRM) codes form a family of locally correctable codes~(LCC) in the regime of low query complexities. A PRM code is specified by the alphabet size $q$, the number of variables $m$, and the degree $d$. When $d\leq q-1$, we present a perfectly smooth local decoder to recover a symbol by accessing $γ\leq q$ symbols to the coordinates fall on a line. There are three major parameters considered in LCCs, namely the query complexity, the message length and the code length. This paper shows that PRM codes are shorter than generalized Reed-Muller~(GRM) codes in LCCs. Precisely, given a GRM code over a field of size $q$, there exists a class of shorter codes over a field of size $q-1$, while maintaining the same values on the query complexities and the message lengths.

DOI
10.48550/arxiv.1702.02671

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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=Θ(q)d=Θ({q})d=Θ(q), where qqq is the code alphabet size (in fact, ddd can be as big as q/4q/4q/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 kkk of coordinates of a Reed-Muller code simultaneously at the cost of querying O(q2k)O(q^2k)O(q2k) coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing kkk locations is in fact cheaper than repeating the procedure for accessing a single location for kkk times. Our estimation of success error probability is based on error probability bound for ttt-wise linearly independent variables given in \cite{BR94}.
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