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Long quantum codes using projective Reed–Muller codes are constructed. Projective Reed–Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed–Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed–Muller codes.

In this work, skew quasi cyclic codes over $R={𝔽}_q+v{𝔽}_q$, where $v^2=v$ are considered. The generating set for one generator skew quasi cyclic codes over $R$ is also determined. We discuss a sufficient condition for one generator skew quasi cyclic codes to be free. Furthermore, a BCH type bound is given for free one generator skew quasi cyclic codes. We investigate the dual of skew quasi cyclic codes over $R$. We give a necessary and sufficient condition for skew cyclic codes over $R$ to contain its dual. Moreover, we construct quantum codes from skew cyclic codes over $R$. By using computer search we give some examples about skew quasi cyclic codes and list some quantum parameters in the table.

Let $p$ be an odd prime, and $k$ be an integer such that $gcd(k,p)=1$. Using pairwise orthogonal idempotents ${\gamma }_1,{\gamma }_2,{\gamma }_3$ of the ring ${ℛ}={𝔽}_p[u]/〈u^{k+1}-u〉$, with ${\gamma }_1+{\gamma }_2+{\gamma }_3=1$, ${ℛ}$ is decomposed as ${ℛ}={\gamma }_1{ℛ}\oplus {\gamma }_2{ℛ}\oplus {\gamma }_3{ℛ}$, which contains the ring $R={\gamma }_1{𝔽}_p\oplus {\gamma }_2{𝔽}_p\oplus {\gamma }_3{𝔽}_p$ as a subring. It is shown that, for ${\lambda }_0,{\lambda }_k\in {𝔽}_p$, ${\lambda }_0+u^k{\lambda }_k\in R$, and it is invertible if and only if ${\lambda }_0$ and ${\lambda }_0+{\lambda }_k$ are units of ${𝔽}_p$. In such cases, we study $({\lambda }_0+u^k{\lambda }_k)$-constacyclic codes over $R$. We present a direct sum decomposition of $({\lambda }_0+u^k{\lambda }_k)$-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a $({\lambda }_0+u^k{\lambda }_k)$-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over ${𝔽}_p$, with better parameters than existing ones, are constructed from cyclic and negacyclic codes over $R$.

CSS codes are a subfamily of stabilizer codes especially appropriate for fault-tolerant quantum computations. A very simple method is proposed to encode a general qudit when a Calderbank–Shor–Steane quantum code, defined over a q-ary alphabet, is used.

We construct new families of duadic codes and derive from them degenerate quantum codes. This provides a solution to a question raised by Aly, Klappenecker and Sarvepalli on the existence of quantum codes from duadic codes when the multiplicative order of q modulo n is even. We also characterize the affine-invariant maximal extended cyclic codes. Then by the CSS construction, we give a family of quantum code.

We give a construction for quantum codes from linear and cyclic codes over . We derive Hermitian self-orthogonal codes over as Gray images of linear and cyclic codes over . In particular, we use two binary codes associated with a cyclic code over of odd length to determine the parameters of the corresponding quantum code.

We introduce a new type of sparse CSS quantum error correcting code based on the homology of hypermaps. Sparse quantum error correcting codes are of interest in the building of quantum computers due to their ease of implementation and the possibility of developing fast decoders for them. Codes based on the homology of embeddings of graphs, such as Kitaev's toric code, have been discussed widely in the literature and our class of codes generalize these. We use embedded hypergraphs, which are a generalization of graphs that can have edges connected to more than two vertices.

We develop theorems and examples of our hypermap-homology codes, especially in the case that we choose a special type of basis in our homology chain complex. In particular the most straightforward generalization of the m × m toric code to hypermap-homology codes gives us a [(3/2)m², 2, m] code as compared to the toric code which is a [2m², 2, m] code. Thus we can protect the same amount of quantum information, with the same error-correcting capability, using less physical qubits.

Constacyclic codes are important classes of linear codes that have been applied to the construction of quantum codes. Six new families of asymmetric quantum codes derived from constacyclic codes are constructed in this paper. Moreover, the constructed asymmetric quantum codes are optimal and different from the codes available in the literature.

In this paper, we construct two classes of new quantum maximum-distance-separable (MDS) codes with parameters , where q is an odd prime power with q ≡ 3 (mod 4) and ; [[8(q - 1), 8(q - 1) - 2d + 2, d]]_q, where q is an odd prime power with the form q = 8t - 1 (t is an even positive integer) and . Comparing the parameters with all known quantum MDS codes, the quantum MDS codes exhibited here have minimum distances bigger than the ones available in the literature.

Let R = F₃ + vF₃ be a finite commutative ring, where v² = 1. It is a finite semi-local ring, not a chain ring. In this paper, we give a construction for quantum codes from cyclic codes over R. We derive self-orthogonal codes over F₃ as Gray images of linear and cyclic codes over R. In particular, we use two codes associated with a cyclic code over R of arbitrary length to determine the parameters of the corresponding quantum code.

In this paper, quantum codes from cyclic codes over A₂ = F₂ + uF₂ + vF₂ + uvF₂, u² = u, v² = v, uv = vu, for arbitrary length n have been constructed. It is shown that if C is self orthogonal over A₂, then so is Ψ(C), where Ψ is a Gray map. A necessary and sufficient condition for cyclic codes over A₂ that contains its dual has also been given. Finally, the parameters of quantum error correcting codes are obtained from cyclic codes over A₂.

We give a construction of quantum codes over ${𝔽}_q$ from cyclic codes over a finite non-chain ring ${𝔽}_q+v{𝔽}_q+v^2{𝔽}_q+v^3{𝔽}_q$, where $q=p^r$, p is a prime, $3|(p-1)$ and $v^4=v$.

In this paper, we study the structure of cyclic, quasi-cyclic codes and their skew codes over the finite ring $B_r=F_2+v_1{F}_2+\cdots +v_r{F}_2$, $v_i^2=v_i,v_i{v}_j=v_j{v}_i=0,1\le i,j\le r$ for $r\ge 1$. The Gray images of cyclic, quasi-cyclic, skew cyclic, skew quasi-cyclic codes over $B_r$ are obtained. A necessary and sufficient condition for cyclic code over $B_r$ that contains its dual has been given. The parameters of quantum error correcting codes are obtained from cyclic codes over $B_r$.

Let $q$ be an odd prime power and $m$ be a positive integer. Maximum designed distance such that negacyclic BCH codes over ${𝔽}_{q^2}$ of length $n=\frac{q^{2m}-1}{2}$ are Hermitian dual-containing codes is given. The dimension of such Hermitian dual-containing negacyclic codes is completely determined by analyzing cyclotomic cosets. Quantum negacyclic BCH codes of length $n=\frac{q^{2m}-1}{2}$ are obtained by using Hermitian construction. The constructed quantum negacyclic BCH codes produce new quantum codes with parameters better than those obtained from quantum BCH codes.

In this paper, we show how to construct hybrid quantum-classical codes from subsystem codes by encoding the classical information into the gauge qudits using gauge fixing. Unlike previous work on hybrid codes, we allow for two separate minimum distances, one for the quantum information and one for the classical information. We give an explicit construction of hybrid codes from two classical linear codes using Bacon–Casaccino subsystem codes, as well as several new examples of good hybrid code.

In the paper, we consider a finite non-chain commutative ring $R={𝔽}_2+u{𝔽}_2+u^2{𝔽}_2$, where $u^3=u$. We mainly study the construction of quantum codes from cyclic codes over $R$. For this, we obtained self-orthogonal codes over ${𝔽}_2$ as gray images of linear and cyclic codes over $R$, then these codes over ${𝔽}_2$ correspond to a cyclic code over $R$ of odd length $n$ used to determine the parameters of the quantum codes.

Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric n-vertices colorable graph and a group of operations (coloring rules) on the graph: find the minimum sequence of operations that maps between two given graph colorings. We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code. Computing the distance of a quantum code is a highly non-trivial problem and our method may be of use in the construction of better codes.

Some results are generalized on linear codes over $Z_3[v]/〈v^3-v〉$ in [15] to the ring $R_p=Z_p[v]/〈v^p-v〉$, where $p$ is an odd prime number. The Gray images of cyclic and quasi-cyclic codes over $R_p$ are obtained. The parameters of quantum error correcting codes are obtained from negacyclic codes over $R_p$. A nontrivial automorphism ${\theta }_p$ on the ring $R_p$ is determined. By using this, the skew cyclic, skew quasi-cyclic, skew constacyclic codes over $R_p$ are introduced. The number of distinct skew cyclic codes over $R_p$ is given. The Gray images of skew codes over $R_p$ are obtained. The quasi-constacyclic and skew quasi-constacyclic codes over $R_p$ are introduced. MacWilliams identities of linear codes over $R_p$ are given.

Additive codes received much attention due to their connections with quantum codes. On the other hand, skew cyclic codes proved to be a useful class of codes that contain many good codes. In this work, we introduce and study additive skew cyclic codes over the quaternary field $\mathrm{\text{GF}}(4)$, obtaining some structural properties of these codes. Moreover, we also show that many best known and optimal quantum codes can be obtained from this class.

In this paper, we study cyclic codes over $R={𝔽}_4+u{𝔽}_4,u^2=0$. A necessary and sufficient condition for a cyclic code over $R$ to contain its dual is determined. The odd and even length cases are discussed separately to obtain above condition. It is shown that Gray image of a cyclic code over $R$ containing its dual is a linear code over ${𝔽}_4$ which also contains its dual. We have then obtained the parameters of corresponding CSS-quantum codes over ${𝔽}_4$. By augmentation, we construct codes with dual-containing property from codes of smaller size containing their duals. Through this construction, we have obtained some optimal quantum codes over ${𝔽}_4$. Some examples have been given to illustrate the results.