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Analele Universităţii "Ovidius" Constanţa. Seria Matematică

About on the obtaining of some quantum code parameters from ( − 2 v m −1 + 1)-constacyclic codes over the ring R = 𝔽 q + v 𝔽 q + · · · + v m −1 𝔽 q

Gökhan Gökgöz, Figen Öke

Published Jul 1, 2026• 0 citations

Abstract

Abstract In this study, the hull dimensions of ( − 2 v m −1 +1)-constacyclic codes over the ring R = 𝔽 q + v 𝔽 q + · · · + v m −1 𝔽 q with v m = v are determined. In addition, by using a special Gray map, a necessary condition is given under which the generator polynomial of a ( − 2 v m −1 + 1)-constacyclic code over R yields, through the polynomial representation of the Gray map, the generator polynomial of the corresponding cyclic code over F q . Later, an example is presented showing that the parameters of a quantum error-correcting code over F q can be obtained from a ( − 2 v m −1 +1)-constacyclic code over R by means of the X-construction method. Furthermore, the case in which the number of ( − 2 v m −1 + 1)-constacyclic codes of length n over R equals the number of cyclic codes of length 2 n over F q is examined, and an example is provided in which the parameters of the quantum error-correcting codes arising from those cyclic codes—which are Gray images of ( − 2 v m −1 + 1)-constacyclic codes over R —are presented.

DOI: 10.2478/auom-2026-0020

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Abstract In this study, the hull dimensions of ( − 2 v m −1 +1)-constacyclic codes over the ring R = 𝔽 q + v 𝔽 q + · · · + v m −1 𝔽 q with v m = v are determined. In addition, by using a special Gray map, a necessary condition is given under which the generator polynomial of a ( − 2 v m −1 + 1)-constacyclic code over R yields, through the polynomial representation of the Gray map, the generator polynomial of the corresponding cyclic code over F q . Later, an example is presented showing that the parameters of a quantum error-correcting code over F q can be obtained from a ( − 2 v m −1 +1)-constacyclic code over R by means of the X-construction method. Furthermore, the case in which the number of ( − 2 v m −1 + 1)-constacyclic codes of length n over R equals the number of cyclic codes of length 2 n over F q is examined, and an example is provided in which the parameters of the quantum error-correcting codes arising from those cyclic codes—which are Gray images of ( − 2 v m −1 + 1)-constacyclic codes over R —are presented.

Let R be the ring 𝔽2m + u𝔽2m, where u2 = 0. We introduce a Gray map from R to 𝔽22m and study (1 + u)-constacyclic codes over R. It is proved that the image of a (1 + u)-constacyclic code length n over R under the Gray map is a distance-invariant binary quasicyclic code of index m and length 2mn. We also prove that every code of length 2mn which is the Gray image of cyclic codes over R of length n is permutation equivalent to a binary quasi-cyclic code of index m. Furthermore, a family of quantum error-correcting codes obtained from the Calderbank-Shor-Steane (CSS) construction applied to (1 + u)-constacyclic codes over R.

We study the algebraic structures of constacyclic codes over a new finite non-chain ring R and their l-Galois dual codes based on a new Gray map ϕ and determine the dimensions of the l-Galois hull of constacyclic codes over R. Furthermore, we propose a quantum construction X of the Galois inner product to construct quantum codes by combining the Calderbank–Shor–Steane (CSS) construction, Hermitian construction and construction X. As an application, by calculating the dimensions of the l-Galois hull of constacyclic codes over R, we obtain some new quantum maximum distance separable (QMDS) codes and new quantum error-correcting codes (QECCs) that are better than the best known ones.

This paper is concerned with, structural properties and construction of quantum codes over Z p by using (1+(p−2)ν)-Constacyclic codes over the finite commutative non-chain ring ℜ = Z p +νZ p where ν 2 = ν and Z p is field having p elements with characteristic p where p is prime. A Gray map is defined between ℜ and Z p 2 . The parameters of quantum codes over Z p are obtained by decomposing (1+(p−2)ν)-constacyclic codes into cyclic and negacyclic codes over Z p . As an application, some examples of quantum codes of arbitrary length, are also obtained.

Hulls and sums of separable constacyclic codes over

\mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)

and new quantum codes

We establish the generator polynomials of the Euclidean and Hermitian duals of separable constacyclic codes over

\mathcal{S} = \mathbb{F}_q \times (\mathbb{F}_q+v\mathbb{F}_q)

, with

q

an odd prime power and

v^2=v

, and we derive the generator polynomials of their Gray images, respectively. The generator polynomials of the Euclidean hulls and Hermitian hulls of separable constacyclic codes over

\mathcal{S}

and their Gray images are presented, respectively. Furthermore, we provide the generator polynomials of the Euclidean sums and Hermitian sums of separable constacyclic codes and their Gray images, respectively. Finally, we propose two methods to yield quantum error-correcting codes (QECCs) from the hulls and sums of separable constacyclic codes over

\mathcal{S}

, and generate new QECCs that outperform the existing ones in terms of parameters.

This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.

Recently, codes over some special ﬁnite rings especially chain rings have been studied. More recently, codes over ﬁnite non-chain rings have been also considered. Study on codes over such rings or rings in general is motivated by the existence of some special maps called Gray maps whose images give codes over ﬁelds. Quantum error-correcting (QEC) codes play a crucial role in protecting quantum information. The construction of quantum codes via classical codes over 2 F was ﬁrst introduced by Calderbank and Shor [4] and Steane [13] in 1996. This method, known as CSS construction, has received a lot of attention and it has allowed to ﬁnd many good quantum stabilizer codes. Later, construction of quantum codes over larger alphabets from classical linear codes over q F has shown by Ketkar et al. in [10]. One direction of the main research in quantum error correction codes is constructing quantum codes that have large minimum distances [9] for a given size and length. In [14], based on classical quaternary constacyclic linear codes, some parameters for quantum codes are obtained. In [8, 9], respectively based on classical negacyclic and constacyclic linear codes some parameters for quantum MDS codes are presented. In this work, we determine self-dual and self-orthogonal codes arising from constacyclic codes over the group ring(Zq[v]/G