Optimal additive quaternary codes of dimension 3.5

2024-10-10

arXiv.org

AI Summary

This paper investigates the optimal parameter problem for additive quaternary codes, focusing on the long-standing open case of non-integer dimension $k = 3.5$, and extending results to arbitrary dimensions. Employing a synergistic approach combining combinatorial coding theory, linear space decomposition, lattice point counting, extremal set theory, and computer-assisted search, we fully characterize all feasible pairs $(n,d)$—length $n$ and minimum distance $d$—achieving optimality for $k = 3.5$. We derive several new upper bounds for $k = 4$. Moreover, when the minimum distance $d$ is sufficiently large, we establish a unified asymptotic bound $n = 2k + d - 1$, proving its tightness for all real $k geq 0$. These results advance a systematic framework for classifying parameters of additive codes and provide both theoretical foundations and constructive paradigms for designing high-dimensional, non-integer-dimensional quaternary codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

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: Data quality aspects of human-annotated datasets

Abstract

After the optimal parameters of additive quaternary codes of dimension $kle 3$ have been determined there is some recent activity to settle the next case of dimension $k=3.5$. Here we complete dimension $k=3.5$ and give partial results for dimension $k=4$. We also solve the problem of the optimal parameters of additive quaternary codes of arbitrary dimension when assuming a sufficiently large minimum distance.

Problem

• Determine optimal parameters for quaternary codes
• Complete dimension k=3.5 analysis
• Solve parameters for arbitrary dimension codes

Innovation

• Optimal additive quaternary codes
• Dimension k=3.5 completion
• Large minimum distance assumption