论文标题
随机集和应用程序的消失空间
Vanishing Spaces of Random Sets and Applications to Reed-Muller Codes
论文作者
论文摘要
我们在$ \ mathbb {f} _2^m $中研究以下自然问题：给定一组$ k $点$ z = \ {z_1，z_1，z_2，\ dots，z_k \} \ subseteq \ subseteq \ subseteq \ mathbb {f} _2 _2^m $，$ $ $ $ $ $ $ $ $ $ $ z $的所有点？ 我们表明，对于$ r \ r \ leqγm$（其中$γ> 0 $是一个小的，绝对常数的）和$ k =（1-ε）\ cdot \ binom {m} {m} {\ leq r} $，对于任何常数$ε> 0 $，最多是$ r $ r $ r $ r $ r $ r $ r $ r $ r $ z z = z = z = z = z = z_k \} $具有尺寸恰好$ \ binom {m} {\ leq r} - k $，概率$ 1- o（1）$。该限制显示，与$ \ binom {\ binom {\ leq r} - \ leq r} - \ binom {\ binom {\ binom {\ binom {\ binom {\ binom {\ gg { \ binom {m} {\ leq r} -k $。 使用此限制，我们表明高度芦苇毛刺代码（$ \ text {rm}（m，d）$，带有$ d>（1-γ）m $）在二进制擦除渠道下“实现容量”，因为对于任何$ε> 0 $，我们可以从$> 0 $中恢复$（1-ε）\ cdot \ cdot \ cdot \ cdot \ binom {m} \ eq m} $ 1- o（1）$。这也意味着$ \ text {rm}（m，d）$也可以从$ \ of biinom {m} {\ leq m-（d/2）} $随机错误中有效地解码。
We study the following natural question on random sets of points in $\mathbb{F}_2^m$: Given a random set of $k$ points $Z=\{z_1, z_2, \dots, z_k\} \subseteq \mathbb{F}_2^m$, what is the dimension of the space of degree at most $r$ multilinear polynomials that vanish on all points in $Z$? We show that, for $r \leq γm$ (where $γ> 0$ is a small, absolute constant) and $k = (1-ε) \cdot \binom{m}{\leq r}$ for any constant $ε> 0$, the space of degree at most $r$ multilinear polynomials vanishing on a random set $Z = \{z_1,\ldots, z_k\}$ has dimension exactly $\binom{m}{\leq r} - k$ with probability $1 - o(1)$. This bound shows that random sets have a much smaller space of degree at most $r$ multilinear polynomials vanishing on them, compared to the worst-case bound (due to Wei (IEEE Trans. Inform. Theory, 1991)) of $\binom{m}{\leq r} - \binom{\log_2 k}{\leq r} \gg \binom{m}{\leq r} - k$. Using this bound, we show that high-degree Reed-Muller codes ($\text{RM}(m,d)$ with $d > (1-γ) m$) "achieve capacity" under the Binary Erasure Channel in the sense that, for any $ε> 0$, we can recover from $(1 - ε) \cdot \binom{m}{\leq m-d-1}$ random erasures with probability $1 - o(1)$. This also implies that $\text{RM}(m,d)$ is also efficiently decodable from $\approx \binom{m}{\leq m-(d/2)}$ random errors for the same range of parameters.
