Projective tilings and full-rank perfect codes

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 full-rank. A tile is called projective if it is a union of one-dimensional subspaces, and the tiling is projective (semi-projective) if both tiles (at least $V$) are projective. Each tiling with projective $V$ of full rank corresponds to a $1$-perfect code of length $|V|/(q-1)$. Moreover, if $U$ is full-rank, then the code is also of full rank. Adapting a known construction [Szabo] full-rank tilings, one can construct for any $q>2$ semi-projective full-rank tilings in $6$-dimensional space and projective full rank tilings in $10$-dimensional space. In particular, there is a full-rank 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 full-rank codes with a lower kernel dimension, $6$, $5$, $4$, $3$.

References

1. Sándor Szabó, “Constructions Related to the Rédei Property of Groups”, J. London Math. Soc., II Ser., 73:3 (2006), 701–715      
2. Sterre den Breeijen, Tilings Of Additive Groups, Master’s Thesis, Radboud University Nijmegen, 2018 https://www.math.ru.nl/~bosma/Students/SterredenBreeijenMSc.pdf
3. Sándor Szabó, “Full-Rank Factorings of Elementary p-Groups by $Z$-Subsets”, Indagationes Mathematicae, 24:4 (2013), 988–995