Projective tilings and full-rank perfect codes

TL;DR

This paper constructs full-rank tilings with projective components in finite vector spaces, leading to the creation of a ternary 1-perfect code of length 13, advancing the understanding of perfect codes and tilings.

Contribution

It introduces new methods for constructing full-rank tilings with projective components, including the first known ternary 1-perfect code of length 13.

Findings

Constructed full-rank tilings with projective components for vector spaces of dimension ≥6.

Provided the first known ternary 1-perfect code of length 13.

Connected tilings with projective components to factorizations of projective spaces.

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. Keywords: perfect codes, tilings, group factorization, full-rank tilings, projective geometry