Subspace coverings with multiplicities
Anurag Bishnoi, Simona Boyadzhiyska, Shagnik Das, Tam\'as, M\'esz\'aros

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 of affine subspaces of codimension that are required to cover all points of at least times while covering the origin at most times. The case is a classic result of Jamison, which was independently obtained by Brouwer and Schrijver for . The value of 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 we have , while for we have , and also study the transition between these two ranges. While previous work in this direction…
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Taxonomy
TopicsCoding theory and cryptography · Limits and Structures in Graph Theory · graph theory and CDMA systems
