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Reed–Muller Codes for Random Erasures and Errors

机译：里德-穆勒码用于随机删节和错误

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This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF(2) polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity. The proofs of these four results are based on dierent techniques, which we nd interesting in their own right. In particular, we study the following questions about E(m; r), the matrix whose rows are truth tables of all monomials of degree r in m variables. What is the most (resp. least) number of random columns in E(m; r) that dene a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes. Our decoding from random errors follows from the following novel reduction. For every linear code C of suciently high rate we construct a new code C0 obtained by tensorizing C, such that for every subset S of coordinates, if C can recover from erasures in S, then C0 can recover from errors in S. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate. Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent [KLP12] bounds from constant degree to linear degree polynomials.
机译：本文研究了可以在二进制擦除信道和二进制对称信道上成功解码二进制Reed-Muller（RM）码的参数，尤其是它们何时可以达到这两个经典信道的容量。必要地，本文还研究了随机输入集上的多元GF（2）多项式的求值性质。为了进行擦除，我们证明了RM代码可以在非常高的速率和非常低的速率范围内实现容量。对于错误，我们证明RM代码可以在非常低的速率范围内实现容量，而对于非常高的速率，则表明它们可以在大约错误数量的平方根下进行唯一解码。这四个结果的证明是基于不同的技术，我们本身很有趣。特别是，我们研究以下关于E（m; r）的问题，E（m; r）的行是m个变量中所有r次单项式的真值表。在E（m; r）中最多（最多）随机列确定具有全列等级（分别为全行等级）的子矩阵的概率是多少？我们获得了非常小（分别非常大）度r的严格边界，我们用它来证明RM代码在这些情况下具有擦除的能力。我们对随机错误的解码来自以下新颖的归约。对于每个速率很高的线性代码C，我们构造一个通过张紧C所获得的新代码C0，这样对于坐标S的每个子集，如果C可以从S中的擦除中恢复，那么C0可以从S中的错误中恢复。 RM代码并将我们的结果用于擦除操作意味着我们的结果是对RM代码进行高速率的独特解码。最后，我们要达到的两个能力要求在RM代码的权重分布上有严格的界限。我们获得了这样的界线，将最近的[KLP12]界线从恒定程度扩展到线性程度多项式。

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