On the covering radius of small codes versus dual distance

TL;DR

This paper investigates the covering radius of small linear codes with dual distance, showing that for subexponential code sizes, almost all points can be covered with a radius close to the theoretical bounds, especially when relaxing coverage requirements.

Contribution

The paper demonstrates that for codes with dual distance o(n), the gap in covering radius bounds can be eliminated by allowing a small fraction of uncovered points, extending results to probability distributions.

Findings

Almost all points can be covered with radius close to the lower bound when d = o(n).

The bounds are tight up to a factor less than 3 compared to random codes.

Results extend to (d-1)-wise independent distributions.

Abstract

Tiet\"{a}v\"{a}inen's upper and lower bounds assert that for block-length-$n$ linear codes with dual distance $d$, the covering radius $R$ is at most $\frac{n}{2}-(\frac{1}{2}-o(1))\sqrt{dn}$ and typically at least $\frac{n}{2}-\Theta(\sqrt{dn\log{\frac{n}{d}}})$. The gap between those bounds on $R -\frac{n}{2}$ is an $\Theta(\sqrt{\log{\frac{n}{d}}})$ factor related to the gap between the worst covering radius given $d$ and the sphere-covering bound. Our focus in this paper is on the case when $d = o(n)$, i.e., when the code size is subexponential and the gap is $w(1)$. We show that up to a constant, the gap can be eliminated by relaxing the covering requirement to allow for missing $o(1)$ fraction of points. Namely, if the dual distance $d = o(n)$, then for sufficiently large $d$, almost all points can be covered with radius $R\leq\frac{n}{2}-\Theta(\sqrt{dn\log{\frac{n}{d}}})$. Compared to random linear codes, our bound on $R-\frac{n}{2}$ is asymptotically tight up to a factor less than $3$. We give applications to dual BCH codes. The proof builds on the author's previous work on the weight distribution of cosets of linear codes, which we simplify in this paper and extend from codes to probability distributions on $\{0,1\}^n$, thus enabling the extension of the above result to $(d-1)$-wise independent distributions.