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Additive codes attaining the Griesmer bound

Sascha Kurz

TL;DR

The paper addresses the problem of achieving optimal length-parameter tradeoffs for additive codes and shows that a Griesmer-type bound can be attained with equality when the minimum distance is large, revealing optimal additive-parameter codes that outperform linear ones.The authors extend the Solomon–Stiffler construction to the additive setting within a Galois-geometric framework, mapping codes to faithful projective h-(n,r,s)_q systems and leveraging incidence/partition techniques to construct large families of codes.Key contributions include a generalized constructive scheme with a parametric representation of the minimum distance $d=\sigma q^{k-1}-\sum \varepsilon_i q^{i-1}$ that yields Griesmer-bound-attaining additive codes, a Smith-normal-form approach to feasibility questions, and extensive small-parameter results across binary, ternary, and higher-field cases demonstrating numerous instances where additive codes outperform linear codes.Together these results deepen the understanding of additive-code performance, provide practical constructions for large-distance regimes, and offer computational and asymptotic tools for exploring optimal additive codes in diverse parameter regimes.

Abstract

Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.

Additive codes attaining the Griesmer bound

TL;DR

The paper addresses the problem of achieving optimal length-parameter tradeoffs for additive codes and shows that a Griesmer-type bound can be attained with equality when the minimum distance is large, revealing optimal additive-parameter codes that outperform linear ones.The authors extend the Solomon–Stiffler construction to the additive setting within a Galois-geometric framework, mapping codes to faithful projective h-(n,r,s)_q systems and leveraging incidence/partition techniques to construct large families of codes.Key contributions include a generalized constructive scheme with a parametric representation of the minimum distance $d=\sigma q^{k-1}-\sum \varepsilon_i q^{i-1}$ that yields Griesmer-bound-attaining additive codes, a Smith-normal-form approach to feasibility questions, and extensive small-parameter results across binary, ternary, and higher-field cases demonstrating numerous instances where additive codes outperform linear codes.Together these results deepen the understanding of additive-code performance, provide practical constructions for large-distance regimes, and offer computational and asymptotic tools for exploring optimal additive codes in diverse parameter regimes.

Abstract

Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.

Paper Structure

This paper contains 22 sections, 150 theorems, 225 equations, 27 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Coding theoretic notation
  4. Geometric notation
  5. Basic constructions and bounds
  6. Asymptotic formulations
  7. A generalization of the Solomon--Stiffler construction
  8. Linear equation systems over $\mathbf{\mathbb{Z}}$ and the Smith normal form
  9. Small parameters
  10. Additive codes over the binary field
  11. Additive codes over the ternary field
  12. Additive codes for dimension 2.5
  13. Generalized type of a faithful projective system
  14. Parameterized series of additive codes that outperform linear codes
  15. Generic results
  16. ...and 7 more sections

Key Result

Lemma 1

(delsarte1972weights) Let $0<w_1<w_2$ be the two non-zero weights of a projective linear code over $\mathbb{F}_q$. Then, there exist positive integers $u$, $t$ such that $w_1=up^t$ and $w_2=(u+1)p^t$, where $p$ is the characteristic of $\mathbb{F}_q$.

Theorems & Definitions (311)

  • Example 1
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Theorem 1
  • Lemma 3
  • Definition 2
  • Lemma 4
  • proof
  • Example 2
  • ...and 301 more