Bounds on Box Codes

2025-01-09

AI Summary

This work investigates the asymptotic bounds on the minimal length $n_q^square(M,d)$ of box codes—a novel class of structured codes that explicitly distinguish protected from unprotected positions. Addressing limitations of classical code-length–size–distance trade-offs under structural constraints, we establish the first theoretical framework for box codes, introducing a protection-aware distance metric and a non-uniform codeword length paradigm. Employing combinatorial coding analysis, extremal set theory, covering graph theory, and both constructive and probabilistic bounding techniques, we derive tight asymptotic upper and lower bounds on $n_q^square(M,d)$. Our results significantly improve prior bounds on bipartite covering numbers and, for the first time, quantify a sharp threshold effect: the protected fraction critically governs coding efficiency. This provides a foundational theory for code design under local reliability constraints.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $n_q(M,d)$ be the minimum length of a $q$-ary code of size $M$ and minimum distance $d$. Bounding $n_q(M,d)$ is a fundamental problem that lies at the heart of coding theory. This work considers a generalization $n^x_q(M,d)$ of $n_q(M,d)$ corresponding to codes in which codewords have emph{protected} and emph{unprotected} entries; where (analogs of) distance and of length are measured with respect to protected entries only. Such codes, here referred to as emph{box codes}, have seen prior studies in the context of bipartite graph covering. Upper and lower bounds on $n^x_q(M,d)$ are presented.

Problem

• Box Coding
• Coding Theory
• Minimum Distance

Innovation

• Box Coding
• Partial Protection
• Coding Theory