Sufficient and necessary conditions for Hermitian (1 + λu)-constacyclic self-orthogonal codes over Fpm +uFpm are obtained, where λ is a unit of Fpm + uFpm. Based on this, a new method for the construction of pm-ary quantum codes from the obtained Hermitian (1+ λu) constacyclic self-orthogonal codes over Fpm+uFpm is given. As an example, [[pm - 1; pm - 2d+ 1, d]]pm quantum MDS codes are constructed for 1 ≤ d ≤ (pm +1)/2 and p ≠ 2, as well as some other quantum codes.
Sufficient and necessary conditions for Hermitian (1 + λu)-constacyclic self-orthogonal codes over Fpm +uFpm are obtained, where λ is a unit of Fpm + uFpm. Based on this, a new method for the construction of pm-ary quantum codes from the obtained Hermitian (1+ λu) constacyclic self-orthogonal codes over Fpm+uFpm is given. As an example, [[pm - 1; pm - 2d+ 1, d]]pm quantum MDS codes are constructed for 1 ≤ d ≤ (pm +1)/2 and p ≠ 2, as well as some other quantum codes.
In this paper quantum codes over Fpm are constructed from self-orthogonal (1 + βu)-constacyclic codes of length N = npsover R = Fpm + uFpm , u2= 0, where p is an odd prime, (n, p) = 1, and β is a non-zero element of Fpm. To obtain such quantum codes, the algebraic structure of (1 + βu)-constacyclic codes of length N over R is studied. The structure of their duals is also determined. From the algebraic structures of the (1+βu)-constacyclic codes and their duals over R, self-orthogonal codes over Fpm are obtained as Gray images of self-orthogonal codes over R. Then applying the CalderbankShor-Steane (CSS) construction to these self-orthogonal codes over Fpm, the parameters of the corresponding quantum codes are determined. An example is given to illustrate the results.
Let $ \mathfrak{R}_{l, k} = {\mathbb F}_{p^m}[u_1, u_2, \cdots, u_k]/ \langle u_{i}^{l} = u_{i}, u_iu_j = u_ju_i = 0 \rangle $, where $ p $ is a prime, $ l $ is a positive integer, $ (l-1)\mid(p-1) $ and $ 1\leq i, j\leq k $. First, we define a Gray map $ \phi_{l, k} $ from $ \mathfrak{R}_{l, k}^n $ to $ {\mathbb F}_{p^m}^{((l-1)k+1)n} $, and study its Gray image. Further, we study the algebraic structure of $ \sigma $-self-orthogonal and $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $, and give the necessary and sufficient conditions for $ \lambda $-constacyclic codes over $ \mathfrak{R}_{l, k} $ to satisfy $ \sigma $-self-orthogonal and $ \sigma $-dual-containing. Finally, we construct quantum codes from $ \sigma $-dual-containing constacyclic codes over $ \mathfrak{R}_{l, k} $ using the CSS construction or Hermitian construction and compare new codes our obtained better than the existing codes in some recent references.
Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. To get q-ary quantum MDS codes, one of the effective ways is to find linear MDS codes C over Fq2satisfying C⊥H⊆ C, where C⊥Hdenotes the Hermitian dual code of C. For a linear code C of length n over Fq2, we say that C is a dual-containing code if C⊥H⊆ C and C≠ Fq2n. Several classes of new quantum MDS codes with relatively large minimum distance have been produced through dual-containing constacyclic MDS codes. These works motivate us to make a careful study on the existence conditions for dual-containing constacyclic codes. We obtain necessary and sufficient conditions for the existence of dual-containing constacyclic codes. Four classes of dual-containing constacyclic MDS codes are constructed and their parameters are computed. Consequently, the quantum MDS codes are derived from these parameters. The quantum MDS codes exhibited here have minimum distance bigger than the ones available in the literature.
In this article, we construct some MDS quantum error-correcting codes (QECCs) from classes of constacyclic codes over Rs= Fp+ u1Fp+ ··· + usFp, ui2= ui, uiuj= ujui= 0, for odd prime p and i, j = 1, 2,⋯, s, i ≠ j. Many QECCs with improved parameters than the existing ones in some of the earlier papers are provided. We present a set of idempotent generators of the ring Rs, and using that we define linear codes, determine all units, and study constacyclic codes over this ring. Among others, we study dual containing constacyclic codes over Rs and construct (non-binary) QECCs. An algorithm to construct QECCs from dual containing constacyclic codes over Rs is obtained that can provide many quantum codes.
For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.
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.