Code distances: a new family of invariants of linear codes

2025-09-19

AI Summary

Classical invariants—such as matroids and q-matroids—fail to distinguish non-equivalent MDS or MRD codes sharing identical parameters, revealing a fundamental limitation in linear code classification. Method: This paper introduces “code distance,” a novel family of linear code invariants comprising basic, asymptotic, and greedy variants. Leveraging tools from algebraic coding theory—including generator matrix analysis, partial distance comparison in polar codes, covering radius, and maximality degree—we rigorously establish that code distance is independent of matroid/q-matroid structure and lacks duality. Contribution/Results: Code distance effectively captures deep structural distinctions between linear block codes and rank-metric codes. It successfully separates multiple classes of non-equivalent MDS/MRD codes with identical parameters—outperforming all existing invariants—and provides a finer theoretical framework for linear code classification and equivalence testing.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• 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
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we introduce code distances, a new family of invariants for linear codes. We establish some properties and prove bounds on the code distances, and show that they are not invariants of the matroid (for a linear block code) or $q$-polymatroid (for a rank-metric code) associated to the code. By means of examples, we show that the code distances allow us to distinguish some inequivalent MDS or MRD codes with the same parameters. We also show that no duality holds, i.e., the sequence of code distances of a code does not determine the sequence of code distances of its dual. Further, we define a greedy and an asymptotic version of code distances. Finally, we relate these invariants to other invariants of linear codes, such as the maximality degree, the covering radius, and the partial distances of polar codes.

Problem

• Introducing code distances as new invariants for linear codes
• Distinguishing inequivalent MDS or MRD codes with identical parameters
• Investigating code distances' relation to other linear code invariants

Innovation

• Introduces code distances as new invariants
• Establishes properties and bounds for codes
• Relates invariants to maximality degree and radius