Optimal codes and arcs for the generalized Hamming weights

2026-01-01

AI Summary

This work investigates the construction of optimal linear codes attaining the Griesmer bound with respect to generalized Hamming weights. By employing finite geometric methods, the authors provide a concise geometric proof of the generalized Griesmer bound and demonstrate that the Solomon–Stiffler-type construction achieves this bound when the minimum distance is sufficiently large. The main contribution lies in the complete characterization of the parameters of optimal binary linear codes of dimension at most 7 and optimal ternary linear codes of dimension at most 5, thereby establishing their optimality in terms of generalized Hamming weights for these low-dimensional cases.

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: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

This text contains some notes on the Griesmer bound. In particular, we give a geometric proof of the Griesmer bound for the generalized weights and show that a Solomon--Stiffler type construction attains it if the minimum distance is sufficiently large. We also determine the parameters of optimal binary codes for dimensions at most seven and the optimal ternary codes for dimensions at most five.

Problem

• generalized Hamming weights
• Griesmer bound
• optimal codes
• binary codes
• ternary codes

Innovation

• generalized Hamming weights
• Griesmer bound
• geometric proof
• optimal codes
• Solomon–Stiffler construction