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Projective tilings and full-rank perfect codes

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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.

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Data availability

The dataset generated during the current study (examples of full-rank ternary \((13,3^{10},3)_3\) perfect codes) is available in the IEEE DataPort repository [13].

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  1. Sobolev Institute of Mathematics, Novosibirsk, Russia, 630090

    Denis S. Krotov

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Krotov, D.S. Projective tilings and full-rank perfect codes. Des. Codes Cryptogr. 91, 3293–3303 (2023). https://doi.org/10.1007/s10623-023-01256-y

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