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

The class of &#x2113;-maximum distance separable (&#x2113;-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 <italic>t</italic>-designs. In this paper, for &#x2113; = 1, we completely solve a conjecture recently proposed by Heng <italic>et al</italic>: (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 &#x2113; and &#x2113;&#x2032;, we construct new &#x2113;-MDS codes from known &#x2113;&#x2032;-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 &#x2113;-MDS codes, which generalize, simplify, or improve some known results in the literature. IEEE