Refine constructions for small-radius covering sequences

Derive more precise length computations for the (2^k − 1, 1)- and (2^k, 1)-covering sequences constructed in Sections III and IV, and construct analogous covering sequences for radii 2 and 3, potentially using Preparata codes.

Background

The Hamming-code and self-dual-sequence constructions yield the paper’s strongest covering sequences for small radii, particularly radius one. However, the resulting lengths are not computed as precisely as desired, and analogous constructions for radii two and three are not supplied. The authors specifically suggest Preparata codes, which provide covering codes for these radii, as a possible basis.

References

  1. The (2k - 1, 1)-CS and the (2k, 1)-CS introduced in Sections III and IV are the best covering sequences obtained for small radii. We would like to have a more precise computation on the length of these sequences. We would also like to see similar sequences for radius 2 and radius 3. The Preparata code used to obtain covering codes for radii 2 and 3 [18], [20] might be the ones to use for this purpose.
— Constructions of Covering Sequences and Arrays  (2502.08424 - Chee et al., 12 Feb 2025) in Section VIII, Conclusion and Future Research, Problem 3