On the tightness of Tiet\"av\"ainen's bound for distributions with limited independence

TL;DR

This paper investigates the tightness of Tiet"av"ainen's bound on the covering radius of distributions with limited independence, showing it is asymptotically tight up to a factor of 2 for certain k-wise independent distributions.

Contribution

The authors prove that Tiet"av"ainen's bound is asymptotically tight for k-wise independent distributions when k is up to n^{1/3}/log^2 n, using a new polynomial approximation lemma.

Findings

Existence of k-wise independent distributions with covering radius close to Tiet"av"ainen's bound.

A new lemma on low degree polynomials and expectations under the binomial distribution.

Application of approximation theory tools to problems in coding theory and probability distributions.

Abstract

In 1990, Tiet\"av\"ainen showed that if the only information we know about a linear code is its dual distance $d$, then its covering radius $R$ is at most $\frac{n}{2}-(\frac{1}{2}-o(1))\sqrt{dn}$. While Tiet\"av\"ainen's bound was later improved for large values of $d$, it is still the best known upper bound for small values including the $d = o(n)$ regime. Tiet\"av\"ainen's bound holds also for $(d-1)$-wise independent probability distributions on $\{0,1\}^n$, of which linear codes with dual distance $d$ are special cases. We show that Tiet\"av\"ainen's bound on $R-\frac{n}{2}$ is asymptotically tight up to a factor of $2$ for $k$-wise independent distributions if $k\leq\frac{n^{1/3}}{\log^2{n}}$. Namely, we show that there exists a $k$-wise independent probability distribution $\mu$ on $\{0,1\}^n$ whose covering radius is at least $\frac{n}{2}-\sqrt{kn}$. Our key technical contribution is the following lemma on low degree polynomials, which implies the existence of $\mu$ by linear programming duality. We show that, for sufficiently large $k\leq\frac{n^{1/3}}{\log^2{n}}$ and for each polynomial $f(v)\in {\mathbb R}[v]$ of degree at most $k$, the expected value of $f$ with respect to the binomial distribution cannot be positive if $f(w)\leq 0$ for each integer $w$ such that $|w-n/2|\leq\sqrt{kn}$. The proof uses tools from approximation theory.