Integer sequences that are generalized weights of a linear code
This work provides a theoretical characterization of weight sequences for coding theorists, but it is incremental as it relies on the existence of MDS/MSRD codes.
The paper characterizes which integer sequences can be realized as sequences of generalized weights for linear block codes, rank-metric codes, and sum-rank metric codes, under the assumption that MDS and MSRD codes exist. It also shows that the same sequences appear as greedy weights and characterizes sequences for relative generalized weights.
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.