Name
Abstract | ||
|---|---|---|
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 $r \leq \gamma m$ (where $\gamma > 0$ is a small, absolute constant) and $k = (1-\epsilon) \cdot \binom{m}{\leq r}$ for any constant $\epsilon > 0$, 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 $d > (1-\gamma) m$) "achieve capacity" under the Binary Erasure Channel in the sense that, for any $\epsilon > 0$, we can recover from $(1 - \epsilon) \cdot \binom{m}{\leq m-d-1}$ 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. |
Year | DOI | Venue |
|---|---|---|
2022 | 10.4230/LIPICS.CCC.2022.31 | Computational Complexity Conference (CCC) |
DocType | Citations | PageRank |
Conference | 0 | 0.34 |
References | Authors | |
0 | 4 | |
Name | Order | Citations | PageRank |
|---|---|---|---|
| Siddharth Bhandari | 1 | 0 | 0.34 |
| Prahladh Harsha | 2 | 371 | 32.06 |
| Ramprasad Saptharishi | 3 | 0 | 1.01 |
| Srikanth Srinivasan | 4 | 132 | 21.31 |