Abstract: A tiling of a vector space S is the pair (U, V ) of its subsets such that every vector in S is uniquely represented as the sum of a vector from U and a vector from V . A tiling is connected to a perfect codes if one of the sets, say U, is projective, i.e., the union of one-dimensional subspaces of S. A tiling (U, V ) is full-rank if the affine span of each of U, V is S. For finite non-binary vector spaces of dimension at least 6 (at least 10), we construct full-rank tilings (U, V ) with projective U (both U and V , respectively). In particular, that construction gives a full-rank ternary 1-perfect code of length 13, solving a known problem. We also discuss the treatment of tilings with projective components as factorizations of projective spaces.

Cite: Krotov D.S.
Projective tilings and full-rank perfect codes
Designs, Codes and Cryptography. 2023. V.91. P.3293–3303. DOI: 10.1007/s10623-023-01256-y WOS Scopus OpenAlex

Dates:

Submitted:	Jul 11, 2022
Accepted:	May 27, 2023
Published print:	Jun 13, 2023
Published online:	Jun 13, 2023

Identifiers:

Web of science:	WOS:001005819600001
Scopus:	2-s2.0-85163094966
OpenAlex:	W4380090997

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DB	Citing
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