Projective tilings and full-rank perfect codes
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A tiling of a finite vector space R is the pair (U,V) of its subsets such that U+V=R and |U|· |V|=|R|. 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 R. A tiling (U,V) is full-rank if the affine span of each of U, V is R. For 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, this construction gives a full-rank ternary 1-perfect code of length 13, solving a known problem. We discuss the treatment of tilings with projective components as factorizations of projective spaces.
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