无限族常循环码支持3设计及其在编码理论中的应用
Infinite families of constacyclic codes supporting 3-designs and their applications in coding theory
AI总结：
本文研究了支持无限族3设计的λ-常循环码，推导了其参数和重量分布，并应用于构造纠缠辅助量子误差校正码和局部可恢复码。
摘要：
常循环码在有限域上具有理论重要性，与代数、代数几何、图论、组合设计和数论等领域密切相关。然而，与经典循环码相比，该领域研究仍显有限。本文提供了两个无限族λ-常循环码，支持无限族3设计，推广了[IEEE Trans. Inf. Theory 69(4): 2341-2354, 2023]中的结果。确定了参数和重量分布。此外，研究了其子域子码及其在构造纠缠辅助量子误差校正码（EAQECCs）和局部可恢复码（LRCs）中的应用。值得注意的是，推导出两类具有负或高正净率的最大纠缠EAQECCs。此外，还获得了两类最优距离和最优维数的LRCs。
英文摘要：
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.
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