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

2026-05-06

AI Summary

This work addresses the underexplored connection between constacyclic codes over finite fields and combinatorial 3-designs, along with their applications in quantum and locally recoverable codes. The authors construct two infinite families of λ-constacyclic codes over 𝔽_{q²}, fully determining their parameters and weight distributions. For the first time, they prove that the supports of these codes yield infinite families of 3-designs, thereby generalizing existing results. By employing subfield subcode techniques, they derive entanglement-assisted quantum error-correcting codes (EAQECCs) with maximal entanglement—achieving either highly positive or even negative net rates—as well as locally recoverable codes (LRCs) that are optimal with respect to distance or dimension. These results significantly broaden the applicability of coding theory in combinatorial design, quantum communication, and distributed storage systems.

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 $λ$-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.

Problem

• constacyclic codes
• 3-designs
• finite fields
• coding theory
• combinatorial designs

Innovation

• constacyclic codes
• 3-designs
• EAQECCs
• locally recoverable codes
• subfield subcodes