Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Divisible Codes

Sascha Kurz

TL;DR

This work develops a comprehensive theory of $\Delta$-divisible codes over finite fields, connecting algebraic, geometric, and combinatorial perspectives. It leverages MacWilliams identities and linear programming to derive existence/nonexistence results, characterizes possible lengths via $q^r$-divisible multisets and $S_q(r)$-adic expansions, and builds a suite of constructions (subfields, concatenation, switching) to realize numerous divisible codes. The paper then maps these divisibility properties to a broad spectrum of applications, notably subspace codes, partial spreads, orthogonal arrays, nets, and related combinatorial designs, often yielding new bounds and guiding code constructions. Collectively, the results illuminate deep links between divisibility, geometry of projective spaces, and coding-theoretic optimization, with practical implications for bounds in subspace codes and network design.

Abstract

A linear code over $\mathbb{F}_q$ with the Hamming metric is called $Δ$-divisible if the weights of all codewords are divisible by $Δ$. They have been introduced by Harold Ward a few decades ago. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes. The determination of the possible lengths of projective divisible codes is an interesting and comprehensive challenge.

Divisible Codes

TL;DR

This work develops a comprehensive theory of -divisible codes over finite fields, connecting algebraic, geometric, and combinatorial perspectives. It leverages MacWilliams identities and linear programming to derive existence/nonexistence results, characterizes possible lengths via -divisible multisets and -adic expansions, and builds a suite of constructions (subfields, concatenation, switching) to realize numerous divisible codes. The paper then maps these divisibility properties to a broad spectrum of applications, notably subspace codes, partial spreads, orthogonal arrays, nets, and related combinatorial designs, often yielding new bounds and guiding code constructions. Collectively, the results illuminate deep links between divisibility, geometry of projective spaces, and coding-theoretic optimization, with practical implications for bounds in subspace codes and network design.

Abstract

A linear code over with the Hamming metric is called -divisible if the weights of all codewords are divisible by . They have been introduced by Harold Ward a few decades ago. Applications include subspace codes, partial spreads, vector space partitions, and distance optimal codes. The determination of the possible lengths of projective divisible codes is an interesting and comprehensive challenge.
Paper Structure (34 sections, 87 theorems, 185 equations, 8 tables)

This paper contains 34 sections, 87 theorems, 185 equations, 8 tables.

Table of Contents

  1. Introduction
  2. An introductory application
  3. Preliminaries
  4. The MacWilliams Equations and the Linear Programming Method
  5. Geometric description of linear codes
  6. Basic results for $\Delta$-divisible multisets of points
  7. Lengths of divisible codes
  8. Applications
  9. Constructions for projective $q^r$-divisible codes or multisets of points with bounded maximum point multiplicity
  10. Constructions using subfields
  11. Computer searches
  12. Two-weight codes
  13. Non-existence results for projective $q^r$-divisible codes
  14. Lengths of projective $q^r$-divisible codes
  15. Applications
  16. ...and 19 more sections

Key Result

Theorem 2.6

(MacWilliams Equations, see macwilliams1977theory) For an $[n,k,d]_q$-code $C$ we have for $0\le i\le n$, where are the Krawtchouck polynomials (here $j$ is considered as variable of a polynomial).

Theorems & Definitions (161)

  • Example 1.1
  • Theorem 2.6
  • Example 2.8
  • Example 2.9
  • Example 2.11
  • Example 2.12
  • Example 2.14
  • Example 2.17
  • Theorem 2.18
  • Example 2.19
  • ...and 151 more