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Projective tilings and full-rank perfect codes

Denis S. Krotov

TL;DR

This work investigates tilings of vector spaces and their connection to $1$-perfect codes over finite fields, introducing semiprojective and projective tilings and proving new full-rank constructions. It provides explicit methods to realize full-rank tilings with projective components for various dimensions, including a ternary $1$-perfect code of length $13$, and analyzes how these tilings yield full-rank codes with controlled kernel dimensions. A key contribution is the demonstration that projective tilings correspond to factorizations of projective spaces, enabling a geometric lens on tilings and codes and raising questions about tile sizes and existence of full-rank factorizations. Overall, the results advance the theory of nonbinary perfect codes, expand the catalog of explicit constructions, and connect tiling theory with projective geometry.

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

Projective tilings and full-rank perfect codes

TL;DR

This work investigates tilings of vector spaces and their connection to -perfect codes over finite fields, introducing semiprojective and projective tilings and proving new full-rank constructions. It provides explicit methods to realize full-rank tilings with projective components for various dimensions, including a ternary -perfect code of length , and analyzes how these tilings yield full-rank codes with controlled kernel dimensions. A key contribution is the demonstration that projective tilings correspond to factorizations of projective spaces, enabling a geometric lens on tilings and codes and raising questions about tile sizes and existence of full-rank factorizations. Overall, the results advance the theory of nonbinary perfect codes, expand the catalog of explicit constructions, and connect tiling theory with projective geometry.

Abstract

A tiling of a vector space is the pair of its subsets such that every vector in is uniquely represented as the sum of a vector from and a vector from . A tiling is connected to a perfect codes if one of the sets, say , is projective, i.e., the union of one-dimensional subspaces of . A tiling is full-rank if the affine span of each of , is . For finite non-binary vector spaces of dimension at least (at least ), we construct full-rank tilings with projective (both and , respectively). In particular, that construction gives a full-rank ternary -perfect code of length , 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
Paper Structure (5 sections, 8 theorems, 15 equations)

This paper contains 5 sections, 8 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructing full-rank semiprojective and projective tilings
  4. Full-rank perfect codes
  5. Factorizations in point-line geometries

Key Result

Proposition 1

Let $(U,V)$ be a semiprojective tiling of a vector space $\mathbb{S}$ over a field $\mathbb{F}$. Let $U^*$ be a complete set of mutually non-colinear representatives of $U$. We assume that $U^*$ is finite and denote $N=|U^*|$ (in particular, $N\cdot(|\mathbb{F}|-1)=|U|-1$${\rm)}$. Let the matrix $H$ then where $V_U = V\cap \langle U \rangle$ and $r = \mathrm{rank}(U)$ (if $U$ is full-rank, then

Theorems & Definitions (16)

  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • Theorem 3
  • ...and 6 more