Explicit constructions attaining the optimal generalized covering-code rate

Construct explicit generalized covering codes that attain the optimal asymptotic rate in both the unrestricted setting and the linear setting over finite fields.

Background

The paper determines the optimal asymptotic rate of generalized covering codes whose covering centers have the product form C^t, both without a linearity requirement and, over finite fields, with a linearity requirement. The probabilistic constructions used to prove these rate formulas are non-explicit: the unrestricted result relies on random subsets and Janson's inequality, while the linear result uses random generator matrices, a second-moment argument, and a structured alteration procedure.

The unresolved issue is to replace these probabilistic existence proofs with explicit constructions in each setting while preserving the optimal asymptotic rate. Such constructions would provide effectively describable code families rather than merely establishing that suitable codes exist.