

2021ary quasigroups and related topics
September 17, 2021 11:00–12:30, Novosibirsk, Sobolev Institute of Mathematics, room 135






Projective tilings and fullrank perfect codes
D. S. Krotov^{} 
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Abstract:
A tiling of a finite vector space $R$ over GF$(q)$ is a pair $ (U,V)$ of its subsets (each of which is called a tile) such that $U \cdot V = R$ and $U \cdot V = R$. In the case when each of $U$, $V$ has full rank (that is, its affine span coincides with the entire space), the tiling is called fullrank. A tile is called projective if it is a union of onedimensional subspaces, and the tiling is projective (semiprojective) if both tiles (at least $V$) are projective. Each tiling with projective $V$ of full rank corresponds to a $1$perfect code of length $V/(q1)$. Moreover, if $U$ is fullrank, then the code is also of full rank. Adapting a known construction [Szabo] fullrank tilings, one can construct for any $q>2$ semiprojective fullrank tilings in $6$dimensional space and projective full rank tilings in $10$dimensional space. In particular, there is a fullrank ternary $1$perfect code of length $13$ with the kernel (set of periods) of dimension $7$. By switching from it, it is possible to obtain fullrank codes with a lower kernel dimension, $6$, $5$, $4$, $3$.
References

Sándor Szabó, “Constructions Related to the Rédei Property of Groups”, J. London Math. Soc., II Ser., 73:3 (2006), 701–715

Sterre den Breeijen, Tilings Of Additive Groups, Master’s Thesis, Radboud University Nijmegen, 2018 https://www.math.ru.nl/~bosma/Students/SterredenBreeijenMSc.pdf

Sándor Szabó, “FullRank Factorings of Elementary pGroups by $Z$Subsets”, Indagationes Mathematicae, 24:4 (2013), 988–995

