A distance-free approach to generalized weights

TL;DR

This paper introduces a unified, distance-free framework for defining generalized weights of linear codes using test families, extending classical results and duality to various metrics including sum-rank, Hamming, and rank-metric.

Contribution

It develops a general theory of generalized weights based on test families, unifying and extending existing concepts across multiple metrics and code families.

Findings

Generalized weights are weakly increasing with certain subsequences strictly increasing.

Duality results similar to Wei's Duality Theorem are established.

The framework applies to sum-rank, Hamming, and rank-metric codes, with new insights into their weight properties.

Abstract

We propose a unified theory of generalized weights for linear codes endowed with an arbitrary distance. Instead of relying on supports or anticodes, the weights of a code are defined via the intersections of the code with a chosen family of spaces, which we call a test family. The choice of test family determines the properties of the corresponding generalized weights and the characteristics of the code that they capture. In this general framework, we prove that generalized weights are weakly increasing and that certain subsequences are strictly increasing. We also prove a duality result reminiscent of Wei's Duality Theorem. The corresponding properties of generalized Hamming and rank-metric weights follow from our general results by selecting optimal anticodes as a test family. For sum-rank metric codes, we propose a test family that results in generalized weights that are closely connected to -- but not always the same as -- the usual generalized weights. This choice allows us to extend the known duality results for generalized sum-rank weights to some sum-rank-metric codes with a nonzero Hamming component. Finally, we explore a family of generalized weights obtained by intersecting the underlying code with MDS or MRD codes.