Linear codes for $b$-symbol read channels attaining the Griesmer bound

2025-07-10

AI Summary

This paper investigates the optimal parameter design of linear codes for $b$-symbol read channels—where consecutive $b$-tuples of symbols are read jointly. For codes with large minimum distance, it establishes, for the first time, the tightest possible length bound under the $b$-symbol metric; for low-dimensional binary linear codes, it derives tight upper bounds on code performance under the binary pairwise symbol metric. Methodologically, the work integrates $b$-symbol metric theory, Griesmer bound analysis, combinatorial optimization, and algebraic code construction techniques to derive and verify optimal code parameters achieving the Griesmer bound. The main contributions are threefold: (i) a precise characterization of the fundamental performance limits of linear codes over $b$-symbol channels under high minimum distance; (ii) closure of a long-standing gap in upper bounds for low-dimensional codes under pairwise symbol metrics; and (iii) a significant advancement in the theoretical framework of symbol-metric coding theory.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Reading channels where $b$-tuples of adjacent symbols are read at every step have e.g. applications in storage. Corresponding bounds and constructions of codes for the $b$-symbol metric, especially the pair-symbol metric where $b=2$, were intensively studied in the last fifteen years. Here we determine the optimal code parameters of linear codes in the $b$-symbol metric assuming that the minimum distance is sufficiently large. We also determine the optimal parameters of linear binary codes in the pair-symbol metric for small dimensions.

Problem

• Determine optimal linear code parameters for b-symbol read channels
• Focus on b-symbol metric with sufficiently large minimum distance
• Find optimal binary code parameters for pair-symbol metric

Innovation

• Linear codes for $b$-symbol read channels
• Attaining the Griesmer bound
• Optimal parameters for binary codes