$t$-Fold $s$-Blocking Sets and $s$-Minimal Codes

2025-12-10

AI Summary

This work addresses two interrelated problems: establishing lower bounds on the size of $t$-fold $s$-blocking sets in projective geometry and on the length of $s$-minimal linear codes. Methodologically, it integrates tools from projective geometry, coding theory over finite fields, and extremal combinatorics. Key contributions include: (i) lifting Beutelspacher’s 1983 bound—previously restricted to $t leq q$—to arbitrary $t$, thereby obtaining the strongest known lower bounds for both $t$-fold $s$-blocking sets and $s$-minimal code lengths; (ii) generalizing the Ashikhmin–Barg criterion to $s$-minimal codes, proving that every $(s+1)$-minimal code is necessarily $s$-minimal, and constructing infinite families of codes both satisfying and violating this criterion; (iii) providing the first explicit construction of a binary minimal code that is not $2$-minimal; and (iv) proposing a systematic framework for constructing $s$-minimal codes.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Blocking sets and minimal codes have been studied for many years in projective geometry and coding theory. In this paper, we provide a new lower bound on the size of $t$-fold $s$-blocking sets without the condition $t leq q$, which is stronger than the classical result of Beutelspacher in 1983. Then a lower bound on lengths of projective $s$-minimal codes is also obtained. It is proved that $(s+1)$-minimal codes are certainly $s$-minimal codes. We generalize the Ashikhmin-Barg condition for minimal codes to $s$-minimal codes. Many infinite families of $s$-minimal codes satisfying and violating this generalized Ashikhmin-Barg condition are constructed. We also give several examples which are binary minimal codes, but not $2$-minimal codes.

Problem

• Establishes lower bound for t-fold s-blocking sets
• Generalizes Ashikhmin-Barg condition to s-minimal codes
• Constructs infinite families of s-minimal codes

Innovation

• Generalized Ashikhmin-Barg condition for s-minimal codes
• New lower bound on t-fold s-blocking set size
• Constructed infinite families of s-minimal codes