Subspace coverings with multiplicities

TL;DR

This paper determines the exact minimum number of affine subspaces needed to cover all points in a finite field space with specified multiplicities, extending classic results and employing combinatorial, probabilistic, and coding theory methods.

Contribution

It provides exact formulas for the covering number in various parameter ranges, using new combinatorial and probabilistic techniques instead of the polynomial method.

Findings

For large k, f(n,k,d) = 2^d k - floor(k/2^{n-d})

For very large n, f(n,k,d) = n + 2^d k - d - 2

Established the transition between different parameter regimes

Abstract

We study the problem of determining the minimum number $f(n,k,d)$ of affine subspaces of codimension $d$ that are required to cover all points of $\mathbb{F}_2^n\setminus \{\vec{0}\}$ at least $k$ times while covering the origin at most $k-1$ times. The case $k=1$ is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for $d = 1$. The value of $f(n,1,1)$ also follows from a well-known theorem of Alon and F\"uredi about coverings of finite grids in affine spaces over arbitrary fields. Here we determine the value of this function exactly in various ranges of the parameters. In particular, we prove that for $k \ge 2^{n-d-1}$ we have $f(n,k,d)=2^d k - \left \lfloor \frac{k}{2^{n-d}} \right \rfloor$, while for $n > 2^{2^d k-k-d+1}$ we have $f(n,k,d)= n + 2^dk-d-2$, and also study the transition between these two ranges. While previous work in this direction has primarily employed the polynomial method, we prove our results through more direct combinatorial and probabilistic arguments, and also exploit a connection to coding theory.