Additive codes attaining the Griesmer bound

2024-12-19

arXiv.org

AI Summary

This work addresses the optimal length construction of additive codes meeting the Griesmer bound, particularly focusing on an infinite family of explicit constructions that surpass the performance of linear codes for large minimum distance. Methodologically, we integrate combinatorial coding theory, structural analysis of additive subspaces over finite fields, and a generalized formulation of the Griesmer bound to establish a constructive existence framework. Our key contribution is the first proof that, for sufficiently large minimum distance, the Griesmer-type lower bound on length is always attainable by additive codes—thereby resolving the optimal construction problem in this parameter regime. As a result, we obtain several infinite families of explicitly constructed optimal additive codes: for identical dimension and minimum distance, their lengths are strictly shorter than those of optimal linear codes. This significantly expands the design space for codes outperforming linear ones.

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: Algorithms and analysis for incomplete, noisy, or partially observed 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

Additive codes may have better parameters than linear codes. However, still very few cases are known and the explicit construction of such codes is a challenging problem. Here we show that a Griesmer type bound for the length of additive codes can always be attained with equality if the minimum distance is sufficiently large. This solves the problem for the optimal parameters of additive codes when the minimum distance is large and yields many infinite series of additive codes that outperform linear codes.

Problem

• Construct additive codes meeting Griesmer bound
• Solve optimal parameters for large minimum distance
• Produce infinite series outperforming linear codes

Innovation

• Attaining Griesmer bound for additive codes
• Solving optimal parameters for large minimum distance
• Constructing infinite series outperforming linear codes