Efficient multi-point local decoding of Reed-Muller codes via interleaved codex

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), 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), we are able to locally recover arbitrarily large number k of coordinates of a Reed-Muller code simultaneously with error probability exp (-Ω (k)) at the cost of querying merely O(q2k) coordinates. It turns out that our local decoding of Reed-Muller codes shows (perhaps surprisingly) that accessing k locations is in fact cheaper than repeating the procedure for accessing a single location for k times. Precisely speaking, to get the same success probability by repeating the local decoding algorithm of a single coordinate, one has to query Ω (qk2) coordinates. Thus, the query complexity of our local decoding is smaller for k=Ω (q). If we impose the same query complexity constraint on both algorithm, our local decoding algorithm yields smaller error probability when k=Ω (qq). In addition, our local decoding is efficient, i.e., the decoding complexity is Poly(k,q). Construction of an interleaved codex is based on concatenation of a codex with a multiplication friendly pair, while the main tool to realize codex is based on algebraic function fields (or more precisely, algebraic geometry codes).

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