Quantum codes from a class of constacyclic codes over finite commutative rings

Abstract

Let [Formula: see text] be an odd prime, and [Formula: see text] be an integer such that [Formula: see text]. Using pairwise orthogonal idempotents [Formula: see text] of the ring [Formula: see text], with [Formula: see text], [Formula: see text] is decomposed as [Formula: see text], which contains the ring [Formula: see text] as a subring. It is shown that, for [Formula: see text], [Formula: see text], and it is invertible if and only if [Formula: see text] and [Formula: see text] are units of [Formula: see text]. In such cases, we study [Formula: see text]-constacyclic codes over [Formula: see text]. We present a direct sum decomposition of [Formula: see text]-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a [Formula: see text]-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over [Formula: see text], with better parameters than existing ones, are constructed from cyclic and negacyclic codes over [Formula: see text].

Quantum Codes Obtained From Constacyclic Codes Over a Family of Finite Rings Fp[u₁, u₂, …, us]

“…In Tables 1, we present some QECCs with better parameters from our study of cyclic (Γ = 1) codes over R 1 . Although such codes in Table 1 are better than codes in recent papers [3], [8], [25]- [28], they have low Hamming distance.…”

“…After that, many good QECCs have been constructed by using the cyclic and constacyclic codes over finite fields. Working over finite rings, researchers have constructed QECCs with better parameters [1]- [3], [5], [6], [8], [21]- [23], [25], [26], [28], [32], [33], [36]. Very recently, Bag et al [7] studied the dual containing property of constacyclic and skew constacyclic codes over finite rings and constructed new non binary QECCs from their studies.…”

Good Classical and Quantum Codes from Multi-Twisted Codes

“…They were all obtained using the same method as the codes in the previous table. Codes with the same parameters listed in Table 6 were found using more complex and indirect methods detailed in [21], [22], [23] , [24], [25], [26]. These constructions usually involve considering an extension ring S of a ground ring or field R, constructing a code over the extension ring, then coming back to R using some sort of Gray map.…”

A Study of FqR-Cyclic Codes and Their Applications in Constructing Quantum Codes

“…After that, in 1998, Calderbank et al [17] studied a construction of QECCs from classical linear codes and the complete proof of their existence. Using Calderbank et al [17] concept, several QECCs have been constructed from cyclic codes and their generalizations over finite rings and finite fields [2]- [4], [11], [22].…”