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Generalized ovals, 2.5-dimensional additive codes, and multispreads

Denis S. Krotov, Sascha Kurz

TL;DR

The paper studies additive codes over finite fields through their geometric counterparts, notably generalized ovals and multispreads, to derive bounds and constructions for code parameters. It advances oval-based and subfield constructions to obtain lower bounds on the size of projective systems and translates these into additive code parameters, including $n_q(5,2;2)$ lower bounds via $q^l$-ary ovals and projections. An ILP-based framework with prescribed automorphisms is developed to find new faithful projective systems and to yield explicit generator matrices for small parameters, highlighting the Griesmer-type bounds applicable to additive codes. The work extends to multispreads in $ ext{PG}(4,q)$, delivering complete classifications for $q=2,3$ and substantial results for $q=4,5$, thereby characterizing the parameters of $ ext{F}_4$-linear 64-ary one-weight codes and enriching the connection between projective geometry and additive coding theory. Overall, the results sharpen bounds, provide concrete constructions, and illuminate the deep links between geometric configurations and additive code theory with potential quantum-code applications.

Abstract

We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e.\ projective systems. It is known that the maximum number of $(l-1)$-spaces in $\operatorname{PG}(2,q)$, such that no hyperplane contains three, is given by $q^l+1$ if $q$ is odd. Those geometric objects are called generalized ovals. We show that cardinality $q^l+2$ is possible if we decrease the dimension a bit. We completely determine the minimum possible lengths of additive codes over $\mathbb{F}_9$ of dimension $2.5$ and give improved constructions for other small parameters. As an application, we consider multispreads in $\operatorname{PG}(4,q)$, in particular, completing the characterization of parameters of $\mathbb{F}_4$-linear $64$-ary one-weight codes.

Generalized ovals, 2.5-dimensional additive codes, and multispreads

TL;DR

The paper studies additive codes over finite fields through their geometric counterparts, notably generalized ovals and multispreads, to derive bounds and constructions for code parameters. It advances oval-based and subfield constructions to obtain lower bounds on the size of projective systems and translates these into additive code parameters, including lower bounds via -ary ovals and projections. An ILP-based framework with prescribed automorphisms is developed to find new faithful projective systems and to yield explicit generator matrices for small parameters, highlighting the Griesmer-type bounds applicable to additive codes. The work extends to multispreads in , delivering complete classifications for and substantial results for , thereby characterizing the parameters of -linear 64-ary one-weight codes and enriching the connection between projective geometry and additive coding theory. Overall, the results sharpen bounds, provide concrete constructions, and illuminate the deep links between geometric configurations and additive code theory with potential quantum-code applications.

Abstract

We present constructions and bounds for additive codes over a finite field in terms of their geometric counterpart, i.e.\ projective systems. It is known that the maximum number of -spaces in , such that no hyperplane contains three, is given by if is odd. Those geometric objects are called generalized ovals. We show that cardinality is possible if we decrease the dimension a bit. We completely determine the minimum possible lengths of additive codes over of dimension and give improved constructions for other small parameters. As an application, we consider multispreads in , in particular, completing the characterization of parameters of -linear -ary one-weight codes.

Paper Structure

This paper contains 9 sections, 16 theorems, 38 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. At most two subspaces in a hyperplane -- generalized ovals
  4. Prescribing automorphisms and integer linear programming
  5. Multispreads in PG$(4,q)$
  6. An alternative construction for sets of subspaces with at most two in each hyperplane
  7. Examples of multispreads
  8. Explicit lists of generator matrices for lower bounds for $n_5(5,2;s)$
  9. Explicit lists of generator matrices for lower bounds for $n_4(5,2;s)$

Key Result

Theorem 1

(ball2025griesmer) If $C$ is an additive $[n,r/h,d]_q^h$ code with generator matrix $G$, then $\mathcal{X}_G(C)$ is a projective $h-(n, r, n-d)_q$ system $\mathcal{S}$, and conversely, each projective $h-(n,r, s)_q$ system $\mathcal{S}$ defines an additive $[n,r/h,n-s]_q^h$ code $C$.

Theorems & Definitions (40)

  • Example 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Definition 4
  • Lemma 1
  • Definition 5
  • Example 2
  • Proposition 1
  • ...and 30 more