Improve lower bounds for covering-sequence lengths

Improve the lower bounds on the length of binary (n, R)-covering sequences beyond the lower bounds inherited from (n, R)-covering codes, with improvements substantially larger than one or two units.

Background

The paper focuses primarily on upper bounds for the minimum length L(n, R) of an (n, R)-covering sequence. The lower bounds reported in the paper are those already available for ordinary covering codes, even though the cyclic-sequence requirement may permit stronger bounds. The authors explicitly identify the derivation of materially improved lower bounds as a future research problem.

References

This area is far from being fully explored and we conclude with the following problems for future research. 1) The current work and also all the previous papers that considered this topic, have concentrating on the upper bounds of (n, R)-CSs. The lower bounds mentioned in this paper are the ones used for (n, R)-covering codes. We would like to see improvements in these lower bounds and not just by one or two (something which is not difficult to do).

— Constructions of Covering Sequences and Arrays  (2502.08424 - Chee et al., 12 Feb 2025) in Section VIII, Conclusion and Future Research, Problem 1