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Computer classification of linear codes based on lattice point enumeration and integer linear programming

Sascha Kurz

TL;DR

The paper tackles the classification of linear codes over finite fields with restricted weight sets, a problem tied to partial spreads in Galois geometry and divisibility constraints. It develops a lattice-point enumeration framework augmented by integer linear programming (ILP) to construct or exclude extensions of given codes, including a Phase 0 ILP feasibility step and techniques for handling gaps in the weight spectrum. Key contributions include new enumeration and non-existence results for projective two-weight codes, $\Delta$-divisible codes, and additive $\mathbb{F}_4$-codes, along with practical refinements that significantly reduce search effort. The computational results demonstrate the approach’s effectiveness and provide new bounds and open cases, impacting bounds on partial spreads and related combinatorial structures in finite geometry.

Abstract

Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in Galois Geometry often require a certain divisibility of the occurring weights. In this paper we present an algorithmic framework for the classification of linear codes over finite fields with restricted sets of weights. The underlying algorithms are based on lattice point enumeration and integer linear programming. We present new enumeration and non-existence results for projective two-weight codes, divisible codes, and additive $\mathbb{F}_4$-codes.

Computer classification of linear codes based on lattice point enumeration and integer linear programming

TL;DR

The paper tackles the classification of linear codes over finite fields with restricted weight sets, a problem tied to partial spreads in Galois geometry and divisibility constraints. It develops a lattice-point enumeration framework augmented by integer linear programming (ILP) to construct or exclude extensions of given codes, including a Phase 0 ILP feasibility step and techniques for handling gaps in the weight spectrum. Key contributions include new enumeration and non-existence results for projective two-weight codes, -divisible codes, and additive -codes, along with practical refinements that significantly reduce search effort. The computational results demonstrate the approach’s effectiveness and provide new bounds and open cases, impacting bounds on partial spreads and related combinatorial structures in finite geometry.

Abstract

Linear codes play a central role in coding theory and have applications in several branches of mathematics. For error correction purposes the minimum Hamming distance should be as large as possible. Linear codes related to applications in Galois Geometry often require a certain divisibility of the occurring weights. In this paper we present an algorithmic framework for the classification of linear codes over finite fields with restricted sets of weights. The underlying algorithms are based on lattice point enumeration and integer linear programming. We present new enumeration and non-existence results for projective two-weight codes, divisible codes, and additive -codes.
Paper Structure (6 sections, 8 theorems, 12 equations)

This paper contains 6 sections, 8 theorems, 12 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric representation of linear codes
  4. Extending linear codes using lattice point enumeration
  5. Enhancing the algorithmic approach via integer linear programming computations
  6. Computational results

Key Result

lemma 1

Let $G$ be a systematic generator matrix of an $[n,k]_q$ code $C$ whose non-zero weights are contained in $\{i\Delta\,:a\le i\le b\}\subseteq \mathbb{N}_{\ge 1}$. By $c(P)$ we denote the number of columns of $G$ whose row span equals $P$ for all points $P\in\mathcal{P}_k$ and set $c(\mathbf{0})=r$ f where $e_i$ denotes the $i$th unit vector in $\mathbb{F}_q^{k+1}$. Then, for every systematic gener

Theorems & Definitions (15)

  • lemma 1
  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • lemma 2
  • proof
  • lemma 3
  • ...and 5 more