On ℓ-MDS codes and a conjecture on infinite families of 1-MDS codes

The class of ℓ-maximum distance separable (ℓ-MDS) codes is a generalization of maximum distance separable (MDS) codes that has attracted a lot of attention due to its applications in several areas such as secret sharing schemes, index coding problems, informed source coding problems and combinatorial t -designs. In this paper, for ℓ = 1, we completely solve a conjecture recently proposed by Heng et al : (Discrete Mathematics, 346(10): 113538, 2023) and obtain infinite families of 1-MDS codes with general dimensions holding 2-designs. These later codes are also proved to be optimal locally recoverable codes. For general positive integers ℓ and ℓ′, we construct new ℓ-MDS codes from known ℓ′-MDS codes via some classical propagation rules involving the extended, expurgated, and (u, u+v) constructions. Finally, we study some general results including characterization, weight distributions, and bounds on maximum lengths of ℓ-MDS codes, which generalize, simplify, or improve some known results in the literature.