Infinite families of constacyclic codes supporting 3-designs and their applications in coding theory

TL;DR

This paper introduces two infinite families of constacyclic codes over finite fields that support 3-designs, with complete parameter and weight distribution analysis, and explores their applications in quantum and locally recoverable codes.

Contribution

It presents new infinite families of constacyclic codes supporting 3-designs, extending previous results, and applies them to quantum error correction and locally recoverable codes.

Findings

Complete parameters and weight distributions of the codes are determined.

Constructs maximal entanglement EAQECCs with negative or high positive net rates.

Develops distance-optimal and dimension-optimal LRCs.

Abstract

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 $\lambda$-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.