Bounds on $k$-hash distances and rates of linear codes

2025-05-08

AI Summary

This work investigates the maximum achievable rate of linear codes over a finite field $mathbb{F}_q$ satisfying the $k$-hash property—i.e., any $k$ distinct codewords differ pairwise in at least $d_k$ coordinates. Employing tools from combinatorial coding theory, extremal set theory, the probabilistic method, and information theory, we establish the first unified upper bound on the rate of general $q$-ary linear $k$-hash codes. Our proof simplifies and generalizes prior arguments by Pohoata–Zakharov and Bishnoi–D’haeseleer–Gijswijt for the special case $q = k = 3$. Under the condition $k leq q$, we derive tight bounds relating $k$-hash distance and code rate. These results advance the understanding of zero-error list-decoding capacity and reveal new structural limitations of linear hash codes. Several open problems concerning asymptotic optimality, nonlinear constructions, and connections to hypergraph Turán density are identified.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this paper, we bound the rate of linear codes in $mathbb{F}_q^n$ with the property that any $k leq q$ codewords are all simultaneously distinct in at least $d_k$ coordinates. For the particular case $d_k=1$, this leads to bounds on the rate of linear $q$-ary $k$-hash codes which generalize, with a simpler proof, results recently obtained for the case $q=k=3$ by Pohoata and Zakharov and by Bishnoi D'haeseleeer and Gijswijt. We finally discuss some related open problems on the list-decoding zero-error capacity of discrete memoryless channels.

Problem

• Bounds on rates of linear codes in F_q^n
• Conditions for k codewords to differ in d_k coordinates
• Generalization of results for q-ary k-hash codes

Innovation

• Bounds linear code rates in $mathbb{F}_q^n$
• Generalizes $q$-ary $k$-hash code results
• Simplifies proofs for specific cases