Integer sequences that are generalized weights of a linear code

TL;DR

This paper characterizes which integer sequences can be realized as generalized weights, greedy weights, or relative weights of various types of linear codes, under certain existence assumptions for optimal codes.

Contribution

It provides a comprehensive characterization of integer sequences that correspond to generalized and greedy weights of linear, rank-metric, and sum-rank metric codes, assuming the existence of MDS and MSRD codes.

Findings

Integer sequences can be realized as generalized weights of various codes.

Sequences of greedy weights match those of generalized weights.

Characterization of sequences as relative weights of linear codes.

Abstract

Which integer sequences are sequences of generalized weights of a linear code? In this paper, we answer this question for linear block codes, rank-metric codes, and more generally for sum-rank metric codes. We do so under an existence assumption for MDS and MSRD codes. We also prove that the same integer sequences appear as sequences of greedy weights of linear block codes, rank-metric codes, and sum-rank metric codes. Finally, we characterize the integer sequences which appear as sequences of relative generalized weights (respectively, relative greedy weights) of linear block codes.