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), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously with error probability $\exp (-\Omega (k))$ at the cost of querying merely $O(q^{2}k)$ 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 $\Omega (qk^{2})$ coordinates. Thus, the query complexity of our local decoding is smaller for $k=\Omega (q)$ . If we impose the same query complexity constraint on both algorithm, our local decoding algorithm yields smaller error probability when $k=\Omega (q^{q})$ . In addition, our local decoding is efficient, i.e., the decoding complexity is ${\mathrm{ Poly}}(k,q)$ . 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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通过交错编码对 Reed-Muller 码进行高效的多点本地解码

Reed-Muller 码是最重要的局部可校正码类别之一。目前，Reed-Muller 码的局部解码是基于对直线或二次曲线的解码来恢复单个坐标。要同时恢复多个坐标，最简单的方法是重复本地解码以恢复单个坐标。这种解码算法可能更昂贵，即需要更高的查询复杂度。在本文中，我们关注具有常用参数机制的 Reed-Muller 码，即评估多项式的​​总次数为 $d=\Theta ({q})$ ， 在哪里 $q$ 是代码字母大小（实际上， $d$ 可以大到 $q/4$ 在我们的环境中）。通过引入一种新的codex变体，即interleaved codex（codex的概念已被用于算术秘密共享），我们能够在本地恢复任意大的数字 $千$ 一个 Reed-Muller 代码的坐标同时具有错误概率 $\exp(-\Omega(k))$ 仅以查询为代价 $O(q^{2}k)$ 坐标。事实证明，我们对 Reed-Muller 代码的本地解码显示（也许令人惊讶) 访问 $千$ 位置实际上比重复访问单个位置的过程更便宜 $千$ 次。准确地说，要通过重复单个坐标的局部解码算法来获得相同的成功概率，必须查询 $\Omega (qk^{2})$ 坐标。因此，我们本地解码的查询复杂度较小 $k=\Omega(q)$ . 如果我们对两种算法施加相同的查询复杂度约束，我们的本地解码算法会在以下情况下产生更小的错误概率 $k=\Omega(q^{q})$ . 此外，我们的本地解码是高效的，即解码复杂度为 ${\mathrm{ Poly}}(k,q)$ . 交错编码的构建基于编码与乘法友好对的串联，而实现编码的主要工具是基于代数函数域（或更准确地说，代数几何代码）。