Constacyclic codes over finite fields are of theoretical importance as they are closely related to a number of areas of mathematics such as algebra, algebraic geometry, graph theory, combinatorial designs and number theory. However, the study of constacyclic codes in this context remains limited compared to classical cyclic codes. This paper provides two infinite families of $λ$-constacyclic codes over $\mathbb{F}_{q^2}$ that support infinite families of 3-designs, which generalize the results in [IEEE Trans. Inf. Theory 69(4): 2341-2354, 2023]. The parameters and weight distributions are determined completely. Besides, we study their subfield subcodes and applications on constructing entanglement-assisted quantum error-correcting codes (EAQECCs) and locally recoverable codes (LRCs). It is worthy to mention that two classes of maximal entanglement EAQECCs with a negative or a high positive net rate are derived. Moreover, two classes of distance-optimal and dimension-optimal LRCs are also obtained.

翻译：有限域上的常循环码具有重要的理论意义，因为它们与代数、代数几何、图论、组合设计及数论等多个数学领域紧密相关。然而，与经典循环码相比，常循环码在此方面的研究仍然有限。本文给出了$\mathbb{F}_{q^2}$上两族无限的$\lambda$-常循环码，它们支持无限的3-设计族，推广了文献[IEEE Trans. Inf. Theory 69(4): 2341-2354, 2023]中的结果。我们完全确定了这些码的参数和重量分布。此外，我们研究了它们的子域子码，以及在构建纠缠辅助量子纠错码（EAQECCs）和局部可修复码（LRCs）中的应用。值得提及的是，我们导出了两类具有负或高正净率的极大纠缠EAQECCs。同时，还得到了两类距离最优和维数最优的LRCs。