Affine vector space partitions

Titelangaben

Bamberg, John ; Filmus, Yuval ; Ihringer, Ferdinand ; Kurz, Sascha:
Affine vector space partitions.
2023
Veranstaltung: Workshop Combinatorics in Digital Communication , 19.-21.04.2023 , Eindhoven, Netherlands.
(Veranstaltungsbeitrag: Workshop , Vortrag )

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Abstract

We call a partition of {0,1}^n into subcubes a subcube partition. Subcube partitions have been studied under the various names in the literature, for instance independent sets of clauses, nonoverlapping covers, disjoint tautologies, dividing formulas, coordinate partitions, orthogonal DNF, and disjoint DNF. A subcube partition is irreducible if the only sub-partitions whose union is a subcube are singletons and the entire partition. It is tight if it "mentions" all coordinates. This concept naturally generalizes to {0,...,q-1}^n and affine subspaces of GF(q)^n.

To our knowledge this concept has not been investigated previously except for the special case of so-called homogeneous irreducible affine subspace partitions which go back to Agievich (2008) and occur somewhat naturally in the study of bent functions. Recently, we started the systematic study of affine vector space partitions. Our main concern are extremal constructions, that is partitions with as few or as many subcubes/subspaces as possible. The topic is closely related to MRD codes and divisible codes . even the Grey code makes an appearance. As a highlight we construct an irreducible tight affine vector space partition of GF(q)^7 into q^3 4-spaces for q even using a permutation polynomial of GF(q^3).