Minimal Binary Linear Codes of Dimension n+4 from Partial Spreads and Their Dual Access Structures

2026-08-05

AI Summary

This work addresses the limitation of conventional minimal binary linear codes imposed by the Ashikhmin–Barg condition by proposing a novel construction method based on partial spread geometric structures over finite fields and specially designed Boolean functions. The approach successfully yields minimal binary linear codes of dimension \(n+4\) whose structure explicitly violates the classical minimality criterion, thereby expanding the design boundaries of minimal codes. The study provides a complete characterization of the constructed codes, including their full weight distribution, weight enumerator, and necessary and sufficient conditions for minimality. Furthermore, the dual access structures of these codes are analyzed, highlighting their potential applications in secure multiparty computation and advanced cryptographic communication systems.

Abstract

Minimal linear codes have significant applications in secret sharing schemes, secure multi-party computation, and cryptography. In this paper, we propose a generic construction of a new family of minimal binary linear codes with dimension n+4 from a special class of Boolean functions. By leveraging the geometric properties of partial spreads in finite fields, we determine the explicit weight distribution and weight enumerator of the constructed codes. Furthermore, we derive a necessary and sufficient condition for these codes to be minimal, and establish that the proposed family yields minimal codes that structurally violate the well-known Ashikhmin-Barg condition, making them highly desirable for advanced communication systems.

Problem

• minimal linear codes
• binary linear codes
• partial spreads
• Ashikhmin-Barg condition
• weight distribution

Innovation

• minimal linear codes
• partial spreads
• Boolean functions
• weight distribution
• Ashikhmin-Barg condition