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.