The Maximal Probability that k-wise Independent Bits are All 1

TL;DR

This paper determines the maximum probability that all bits are 1 in a k-wise independent distribution with identical marginals, providing bounds that match existing upper bounds and connecting to error-correcting codes.

Contribution

It introduces explicit bounds for the probability that all bits are 1 in k-wise independent distributions, matching known upper bounds and applying moment problem techniques.

Findings

Derived explicit lower bounds matching upper bounds for the probability

Connected the problem to error-correcting code size limits

Applied moment problem theory to bound polynomial expectations

Abstract

A k-wise independent distribution on n bits is a joint distribution of the bits such that each k of them are independent. In this paper we consider k-wise independent distributions with identical marginals, each bit has probability p to be 1. We address the following question: how high can the probability that all the bits are 1 be, for such a distribution? For a wide range of the parameters n,k and p we find an explicit lower bound for this probability which matches an upper bound given by Benjamini et al., up to multiplicative factors of lower order. The question we investigate can be seen as a relaxation of a major open problem in error-correcting codes theory, namely, how large can a linear error correcting code with given parameters be? The question is a type of discrete moment problem, and our approach is based on showing that bounds obtained from the theory of the classical moment problem provide good approximations for it. The main tool we use is a bound controlling the change in the expectation of a polynomial after small perturbation of its zeros.