Showing papers by "Johan Håstad published in 2016"

An improved bound on the fraction of correctable deletions

Abstract: We consider codes over fixed alphabets against worst case symbol deletions. For any fixed $k \ge 2$ , we construct a family of codes over alphabet of size $k$ with positive rate, which allow efficient recovery from a worst case deletion fraction approaching $1-({2}/({k+\sqrt {k}}))$ . In particular, for binary codes, we are able to recover a fraction of deletions approaching $1/(\sqrt {2} +1)=\sqrt {2}-1 \approx 0.414$ . Previously, even non-constructively, the largest deletion fraction known to be correctable with positive rate was $1-\Theta (1/\sqrt {k})$ , and around 0.17 for the binary case. Our result pins down the largest fraction of correctable deletions for $k$ -ary codes as $1-\Theta (1/k)$ , since $1-1/k$ is an upper bound even for the simpler model of erasures where the locations of the missing symbols are known. Closing the gap between $(\sqrt {2} -1)$ and 1/2 for the limit of worst case deletions correctable by binary codes remains a tantalizing open question.

An Average-Case Depth Hierarchy Theorem for Higher Depth

Abstract: We extend the recent hierarchy results of Rossman, Servedio and Tan [1] to address circuits of almost logarithmic depth. Our proof uses the same basic approach as [1] but a number of small differences enables us to obtain a stronger result by a significantly shorter proof.

An average-case depth hierarchy theorem for higher depth.

Abstract: We extend the recent hierarchy results of Rossman, Servedio and Tan [1] to address circuits of almost logarithmic depth. Our proof uses the same basic approach as [1] but a number of small differences enables us to obtain a stronger result by a significantly shorter proof.

The square lattice shuffle, correction

Abstract: 1. A CORRECTION Theorem 3.6 in [1] contains an error in the stated bound. As is argued above the theorem, the statistical distance to the uniform distribution decreases by a factor O ( (m/ log m)−1/2 ) = O ((log n)1/2n−1/4) for every two iterations. This implies that the bound obtained for Theorem 3.6 should read O ( n1− t−1 2 1 4 (log n) t−1 2 1 2 ) . I am grateful to Justin Holmgren for pointing out this correction.

Bounded Independence vs. Moduli

Abstract: Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.