Generalized ovals, 2.5-dimensional additive codes, and multispreads

2025-11-19

AI Summary

This study addresses the construction and bound problems for generalized elliptic, 2.5-dimensional additive, and multipartition codes over finite fields. Employing finite projective geometry, we establish a geometric correspondence between additive codes and generalized ovals; analyze extremal distributions of low-dimensional subspaces via projection systems; and integrate additive coding theory with algebraic coding techniques for constructive and bounding purposes. Key contributions include: (i) the first construction of $q^l + 2$ $(l-1)$-dimensional subspaces in $mathrm{PG}(l,q)$, breaking the long-standing $q^l + 1$ barrier; (ii) complete determination of the minimum length of 2.5-dimensional additive codes over $mathbb{F}_9$, along with full characterization of their optimal parameters; (iii) improved constructions of additive codes for several small parameter sets; (iv) resolution of existence questions for multipartitions in $mathrm{PG}(4,q)$; and (v) advancement of the parameter classification of $mathbb{F}_4$-linear 64-ary one-weight codes.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• Constructing additive codes using geometric projective systems
• Determining maximum generalized ovals in finite projective spaces
• Characterizing parameters of linear one-weight codes via multispreads

Innovation

• Constructing additive codes via projective systems
• Generalizing ovals to achieve cardinality q^l+2
• Determining minimum lengths for dimension 2.5 codes