The class of $\ell$-maximum distance separable ($\ell$-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 $\ell=1$, 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 been proven to be optimal locally recoverable codes. For general {positive integers} $\ell$ and $\ell'$, we construct new $\ell$-MDS codes from known $\ell'$-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 $\ell$-MDS codes, which generalize, simplify, or improve some known results in the literature.

翻译：$\ell$-最大距离可分 ($\ell$-MDS) 码是最大距离可分 (MDS) 码的一种推广，因其在秘密共享方案、索引编码问题、信息源编码问题以及组合 $t$-设计等多个领域的应用而备受关注。本文针对 $\ell=1$ 的情形，完整解决了 Heng 等人 (Discrete Mathematics, 346(10): 113538, 2023) 近期提出的一个猜想，获得了具有一般维数且承载 $2$-设计的无穷族 $1$-MDS 码。这些码随后被证明是最优局部可修复码。对于一般正整数 $\ell$ 和 $\ell'$，我们通过涉及扩展、收缩及 $(u,u+v)$ 构造等经典传播规则，从已知的 $\ell'$-MDS 码构造成新的 $\ell$-MDS 码。最后，我们研究了一些一般性结果，包括 $\ell$-MDS 码的表征、重量分布以及最大长度界，这些结果推广、简化或改进了文献中的若干已知结论。