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This paper presents a study on the structure of 1-generator quasi-cyclic (QC) codes over the non-chain ring R = F q + u F q + v F q + u v F q , where u 2 = v 2 = 0 , u v = v u , and F q is a finite field of cardinality q = p r ; p is a prime. A minimal spanning set and size of these codes are determined. A sufficient condition for 1-generator QC codes over R to be free is given. BCH-type bounds on the minimum distance of free QC codes over R are also presented. Some optimal linear codes over F q are obtained as the Gray images of quasi-cyclic codes over R. Some characterizations of the Gray images of QC codes over R in F q and F q + u F q (u 2 = 0) are done. As an application, we consider self-orthogonal subcodes of the Gray images of QC codes over R to obtain new and better quantum codes than those are available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study skew constacyclic codes over a class of non-chain rings T = F q [ u 1 , u 2 , ... , u r ] / ⟨ f i (u i) , u i u j - u j u i ⟩ i , j = 1 r , where q = p m , p is some odd prime, m is a positive integer, and f i (u i) are non-constant, monic polynomials that split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T and their dual. We characterize Euclidean and Hermitian dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many maximum distance separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian linear complementary dual (LCD) codes over T , and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous MDS codes and best known linear codes over F q . [ABSTRACT FROM AUTHOR]

In this paper, we provide a method to construct quantum codes from matrix-product codes which the constituent codes have no restrictions. Then, we construct several classes of quantum codes. Some of our quantum codes are new in the sense that the parameters are different from all the previous constructions. In addition, we have that the lengths of some quantum codes are same from the previous constructions, but these quantum codes have larger dimensions. [ABSTRACT FROM AUTHOR]

CSS-T codes were recently introduced as quantum error-correcting codes that respect a transversal gate. A CSS-T code depends on a CSS-T pair, which is a pair of binary codes (C 1 , C 2) such that C 1 contains C 2 , C 2 is even, and the shortening of the dual of C 1 with respect to the support of each codeword of C 2 is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes (C 1 , C 2) is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed–Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature. [ABSTRACT FROM AUTHOR]

This paper presents some results on 1-generator quasi-cyclic (QC) codes and their duals over finite fields. We have explicitly obtained a set of generators for the dual of a 1-generator QC code C from the generator of C, for some restricted cases. From the form of the generators of the dual code C ⊥ , we have derived upper bounds on the minimum distance of C ⊥ . For their applications in constructing quantum codes, we have presented some criteria for 1-generator QC codes to be self-orthogonal or self-dual. Then using the CSS construction, we have obtained some new and better quantum codes with the help of self-orthogonal 1-generator QC codes. [ABSTRACT FROM AUTHOR]

In this paper, we propose a construction of quantum codes from Hermitian self-orthogonal matrix product codes over the finite fields. This construction is applied to obtain numerous new quantum codes, and all of them have higher rate than current quantum codes available. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in right (resp., left) quasi-polycyclic (QP) codes of length n = m ℓ with an associated vector a = (a 0 , a 1 , ... , a m - 1) ∈ F q m , which are a generalization of quasi-cyclic codes (QC) and quasi-twisted (QT) codes. They are defined as invariant subspaces of F q n by the right (resp., left) QP operator T ~ a → (resp., T ~ a ← ). A correspondence between the right (resp., left) ℓ -QP codes and the linear codes of length ℓ over the ring R a → , m : = F q [ x ] / x m - a → (x) , a → (x) = ∑ i = 0 m - 1 a i x i (resp., R a ← , m : = F q [ x ] / x m - a ← (x) , a ← (x) = ∑ i = 0 n - 1 a i x m - 1 - i ) is given. This correspondence leads to some basic characterizations of these codes such as generator and parity check polynomials. Moreover, we also discuss the structure of the 1-generator QP codes, and we prove BCH-like and the Hartmann–Tzeng-like bounds on the minimum distance of QP codes. Finally, we give examples of new quantum codes derived from QP codes as an application of some of the results. Several of these codes are MDS. [ABSTRACT FROM AUTHOR]

In this paper, let q be an odd prime power. Based on new constacyclic codes which contain their Hermitian duals and Hermitian construction, we construct some classes of quantum MDS codes and quantum codes. When q ≡ 1 mod 4 , x and y are a divisor of q - 1 and q + 1 , respectively, we can construct a class of new quantum codes of length n = 2 x y q 2 m - 1 q 2 - 1 for odd x , y , m ≥ 3 . These codes have larger dimensions than existing codes. In addition, for q with the form 2 a m ± (x 2 + y 2) a - 1 and odd x, y, a with g c d (x , y) = 1 , we get some quantum MDS codes of length n = q 2 + 1 a . [ABSTRACT FROM AUTHOR]

In the setting of entanglement-assisted quantum error-correcting codes (EAQECCs), the sender and the receiver have access to noiseless pre-shared entanglement. With this assumption, one may obtain codes with better information rates or improved error handling properties. Entanglement incurs costs and must be judiciously calibrated in designing quantum codes, relative to their deployment parameters. Revisiting known constructions, we devise tools from classical coding theory to better understand how the amount of entanglement can be varied. We present three new propagation rules for the parameters of these codes. Tables listing the parameters of qubit and qutrit EAQECCs that we can explicitly construct are supplied for reference and comparison. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of Hermitian dual-containing constacyclic codes of length n = q 2 m - 1 q + 1 , where q is a power of 2 and m ≥ 2 is an integer. We apply the Hermitian construction to gain quantum constacyclic codes, and lots of codes obtained in the present paper have parameters better than those of the known quantum codes. The Hermitian hull of a linear code is defined to be the intersection of itself and its Hermitian dual. We construct a family of entanglement-assisted quantum error-correcting (EAQEC) codes with n = 4 m - 1 3 via determining the dimensions of the Hermitian Hulls. [ABSTRACT FROM AUTHOR]

Current work presents a new approach to quantum surface codes on compact surfaces with genus g ≥ 2 using the identification of these surfaces with hyperbolic polygons and hyperbolic semi-regular tessellations. This method generalizes other contructions, and we show that this approach may give rise to codes with very good parameters. We present tables with several examples of these codes whose parameters had not been shown before. [ABSTRACT FROM AUTHOR]

It is well-known that it is an important task to construct quantum codes with good parameters. In this paper, we first give the definition of the Euclidean sums of classical codes, and prove that the Euclidean sums of classical codes are Euclidean dual-containing. Then we construct five classes of quantum codes based on the Euclidean sums of the matrix-product codes. We compare our quantum codes with previously known quantum codes available in the literature. As can be seen, some of them have not been constructed before, and in some cases, have larger dimension than previous literature. [ABSTRACT FROM AUTHOR]

In the current paper, we study skew cyclic codes over the ring S : = F 2 × (F 2 + v F 2) with v 2 = v . We investigate the structural properties of skew cyclic codes over the ring S . Also, we describe the dual codes of skew cyclic codes with respect to the Euclidean inner product and the Hermitian inner product. Furthermore, we give a necessary and sufficient condition for a skew cyclic code over S to contain its dual. Using these results, we show that binary quantum error-correcting codes as well as entanglement-assisted quantum error-correcting codes can be constructed from skew cyclic codes over S . Finally, we give examples of binary entanglement-assisted quantum error-correcting codes with good parameters from skew cyclic codes over S . [ABSTRACT FROM AUTHOR]

Let p be a prime integer and m and s be two positive integers. We compute dual codes and Hamming distances of negacyclic codes of length p s over F p m + u F p m with u 2 = 0 . This yields new quantum codes from negacyclic codes, which are neither self-dual nor dual-containing codes. [ABSTRACT FROM AUTHOR]

Weight parity error correction (WPEC) is an error-correction scheme that can correct high-weight Pauli errors. Still, it was initially shown to apply only to Steane and Golay codes. In this work, we prove that the WPEC scheme is appropriate for concatenated codes constructed from two classes of quantum codes, quantum quadratic-residue codes and quantum stabilizer codes from self-orthogonal cyclic codes. We generalize the implementation of the WPEC technique to nonbinary quantum codes and provide an application using WPEC for the [ [ 121 , 1 , 25 ] ] 3 concatenated code. We also show that the WPEC may not be applicable for some quantum codes with specific parameters through an example of the [[15, 7]] code over F 2 . [ABSTRACT FROM AUTHOR]

Due to efficient encoding and decoding algorithms, BCH codes have many applications in data storage systems and satellite communications. Meanwhile, one advantage of BCH codes is to construct quantum codes. In this paper, we assume that q ≡ 1 mod m , where m is a positive integer. We first study a class of q - ary quantum codes of length r (q m - 1) q - 1 by CSS construction, where r ∣ q - 1 . Then, we construct two families of q - ary quantum codes of length r (q 2 m - 1) q 2 - 1 by Hermitian construction. These constructions produce new quantum codes with larger dimensions. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m ≥ 2 be a positive integer. Based on classical cyclic codes, we construct new quantum codes by using images of cyclic codes. Three classes of quantum codes are derived from the F q 2 images of cyclic codes over F q 2 m . Compared with the known ones, these new quantum codes have larger minimum distance or higher code rate. [ABSTRACT FROM AUTHOR]

One of the main objectives of quantum error-correction theory is to construct quantum codes with optimal parameters and properties. In this paper, we propose a class of 2-generator quasi-cyclic codes and study their applications in the construction of quantum codes over small fields. Firstly, some sufficient conditions for these 2-generator quasi-cyclic codes to be dual-containing concerning Hermitian inner product are determined. Then, we utilize these Hermitian dual-containing quasi-cyclic codes to produce quantum codes via the famous Hermitian construction. Moreover, we present a lower bound on the minimum distance of these quasi-cyclic codes, which is helpful to construct quantum codes with larger lengths and dimensions. As the computational results, many new quantum codes that exceed the quantum Gilbert-Varshamov bound are constructed over F q , where q is 2, 3, 4, 5. In particular, 16 binary quantum codes raise the lower bound on the minimum distance in Grassl’s table (Grassl in bounds on the minimum distance of linear codes and quantum codes. ). In nonbinary cases, many quantum codes are new or have better parameters than those in the literature. [ABSTRACT FROM AUTHOR]

Let p be an odd prime, q = p m , R 1 = F q + u F q + v F q + u v F q with u 2 = 1 , v 2 = 1 , u v = v u and R 2 = F q [ u , v , w ] / ⟨ u 2 - 1 , v 2 - 1 , w 2 - 1 , u v - v u , v w - w v , w u - u w ⟩ with u 2 = 1 , v 2 = 1 , w 2 = 1 , u v = v u , v w = w v , w u = u w . In this paper, F q R 1 R 2 -cyclic codes are introduced. We construct quantum error-correcting codes from F q R 1 R 2 -cyclic codes and introduced a Gray map to find new and better quantum error-correcting codes than previously known quantum error-correcting codes over F q . [ABSTRACT FROM AUTHOR]

By recursive construction and computer-supported search method, the spectrum of quantum caps in projective space P G (k , 9) (k ≥ 2) is determined. Through Hermitian construction, the obtained quantum caps are used to construct ternary quantum codes of d = 4 . As a result, for each integer n satisfying n ≥ 10 , a quantum n-cap in P G (k - 1 , 9) with suitable k and its related ternary [ [ n , n - 2 k , 4 ] ] quantum code are constructively proven to exist. According to quantum GV bound and quantum Hamming bound, all quantum codes are optimal. In addition, while constructing quantum caps, the cardinalities of maximal caps in PG(7, 9) and PG(8, 9) are also improved. [ABSTRACT FROM AUTHOR]

In this paper, we propose a construction of q-ary quantum codes from Hermitian self-orthogonal cyclic codes over the finite chain ring R = F q 2 [ u ] / ⟨ u k ⟩ , with u k = 0 , where F q 2 is a finite field with q 2 elements, q = p m , p a prime. A Gray map from R to F q 2 k is defined. Some characterizations of cyclic codes over R have been given in terms of their different types of generators. The structure of their dual codes has also been determined, and a necessary and sufficient condition for these codes to be self-orthogonal is presented. The construction of quantum codes is derived by applying Hermitian construction to the Gray images of self-orthogonal cyclic codes over R. From this construction, we have been able to obtain some new quantum codes with better parameters than some presently known comparable best codes available in the literature. Some of these codes have been given as examples, and the others have been given in the form of two tables. [ABSTRACT FROM AUTHOR]

In this paper, we study a family of constacyclic BCH codes over F q 2 of length n = q 2 m - 1 q + 1 , where q is a prime power, and m ≥ 2 an even integer. The maximum designed distance of narrow-sense Hermitian dual-containing constacyclic BCH codes over F q 2 of length n is determined. Furthermore, the exact dimensions of these constacyclic BCH codes with given designed distance are obtained. As a consequence, we are able to derive the parameters of quantum codes as a function of their designed parameters of the associated constacyclic BCH codes. This improves a recent result by Yuan et al. (Des Codes Cryptogr 85(1): 179–190, 2017) for codes with the same lengths except three trivial cases ( q = 2 , 3 , 4 ). Moreover, some of our newly constructed quantum codes have better parameters compared with those constructed recently (Song et al. Quantum Inf Process 17(10): 1–24, 2018, Aly et al. IEEE Trans Inf Theory 53(3): 1183–1188, 2007, Li et al. Quantum Inf Process 18(5): 127, 2019, Wang et al. Quantum Inf Process 18(10): 1–40, 2019). [ABSTRACT FROM AUTHOR]

Let R be the finite non-chain ring F q 2 + v F q 2 , where v 2 = v and q is an odd prime power. In this paper, we study quantum codes over F q from constacyclic codes over R . We define a class of Gray maps, which preserves the Hermitian dual-containing property of linear codes from R to F q 2 . We study α (1 - 2 v) -constacyclic codes over R , and show that the images of α (1 - 2 v) -constacyclic codes over R under the special Gray map are α 2 -constacyclic codes over F q 2 . Some new non-binary quantum codes are obtained via the Gray map and the Hermitian construction from Hermitian dual-containing α (1 - 2 v) -constacyclic codes over R . [ABSTRACT FROM AUTHOR]

In 2001, Blackmore and Norton introduced an important tool called matrix-product codes, which turn out to be very useful to construct new quantum codes of large lengths. To obtain new and good quantum codes, we first give a general approach to construct matrix-product codes being Hermitian dual-containing and then provide the constructions of such codes in the case s ∣ (q 2 - 1) , where s is the number of the constituent codes in a matrix-product code. For s ∣ (q + 1) , we construct such codes with lengths more flexible than the known ones in the literature. For s ∣ (q 2 - 1) and s ∤ (q + 1) , such codes are constructed in an unusual manner; some of the constituent codes therein are not required to be Hermitian dual-containing. Accordingly, by Hermitian construction, we present two procedures for acquiring quantum codes. Finally, we list some good quantum codes, many of which improve those available in the literature or add new parameters. [ABSTRACT FROM AUTHOR]

In this work, our main objective is to construct quantum codes from quasi-twisted (QT) codes. At first, a necessary and sufficient condition for Hermitian self-orthogonality of QT codes is introduced by virtue of the Chinese remainder theorem. Then, we utilize these self-orthogonal QT codes to provide quantum codes via the famous Hermitian construction. Moreover, we present a new construction method of q-ary quantum codes, which can be viewed as an effective generalization of the Hermitian construction. General QT codes that are not self-orthogonal are also employed to construct quantum codes. As the computational results, some binary, ternary and quaternary quantum codes are constructed and their parameters are determined, which all cannot be deduced by the quantum Gilbert–Varshamov bound. In the binary case, a small number of quantum codes are derived with strictly improved parameters compared with the current records. In the ternary and quaternary cases, our codes fill some gaps or have better performances than the current results. [ABSTRACT FROM AUTHOR]

Bicyclic codes are a generalization of the one-dimensional (1D) cyclic codes to two dimensions (2D). Similar to the 1D case, in some cases, 2D cyclic codes can also be constructed to guarantee a specified minimum distance. Many aspects of these codes are yet unexplored. Motivated by the problem of constructing quantum codes, we study some structural properties of certain bicyclic codes. We show that a primitive narrow-sense bicyclic hyperbolic code of length n 2 contains its dual if and only if its design distance is lower than n - O (n) . We extend the sufficiency condition to the non-primitive case as well. We also show that over quadratic extension fields, a primitive bicyclic hyperbolic code of length n 2 contains Hermitian dual if and only if its design distance is lower than n - O (n) . Our results are analogous to some structural results known for BCH and Reed–Solomon codes. They further our understanding of bicyclic codes. We also give an application of these results by showing that we can construct two classes of quantum bicyclic codes based on our results. [ABSTRACT FROM AUTHOR]

Let q be a prime power and m ≥ 2 be a positive integer. A sufficient condition for the q 2 -ary images of constacyclic codes over F q 2 m to be Hermitian self-orthogonal is presented. Hermitian self-orthogonal codes over F q 2 are obtained as the images of constacyclic codes over F q 2 m . Two classes of quantum codes are derived by employing the Hermitian construction. The construction produces quantum codes with better parameters than the previously known ones. [ABSTRACT FROM AUTHOR]

Let q be a prime power with gcd (q , 6) = 1 . Let R = F q 2 + u F q 2 + v F q 2 + u v F q 2 , where u 2 = u , v 2 = v and u v = v u . In this paper, we give the definition of linear skew constacyclic codes over F q 2 R . By the decomposition method, we study the structural properties and determine the generator polynomials and the minimal generating sets of linear skew constacyclic codes. We define a Gray map from F q 2 α × R β to F q 2 α + 4 β preserving the Hermitian orthogonality, where α and β are positive integers. As an application, by Hermitian construction, we obtain some good quantum error-correcting codes. [ABSTRACT FROM AUTHOR]

Let R be the polynomial residue ring F q 2 + u F q 2 , where F q 2 is the finite field with q 2 elements, q is a power of a prime p, and u is an indeterminate with u 2 = 0. We introduce a Gray map from R to F q 2 p and study (1 - u) -constacyclic codes over R. It is proved that the image of a (1 - u) -constacyclic code of length n over R under the Gray map is a distance-invariant linear cyclic code of length pn over F q 2. We give some necessary and sufficient conditions for (1 - u) -constacyclic codes over R to be Hermitian dual-containing. In particular, a new class of 2 m -ary quantum codes is obtained via the Gray map and the Hermitian construction from Hermitian dual-containing (1 - u) -constacyclic codes over the ring F 2 2 m + u F 2 2 m . [ABSTRACT FROM AUTHOR]

Let $$R=\mathbb {F}_{2^{m}}+u\mathbb {F}_{2^{m}}+\cdots +u^{k}\mathbb {F}_{2^{m}}$$ , where $$\mathbb {F}_{2^{m}}$$ is the finite field with $$2^{m}$$ elements, m is a positive integer, and u is an indeterminate with $$u^{k+1}=0.$$ In this paper, we propose the constructions of two new families of quantum codes obtained from dual-containing cyclic codes of odd length over R. A new Gray map over R is defined, and a sufficient and necessary condition for the existence of dual-containing cyclic codes over R is given. A new family of $$2^{m}$$ -ary quantum codes is obtained via the Gray map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over R. In particular, a new family of binary quantum codes is obtained via the Gray map, the trace map and the Calderbank-Shor-Steane construction from dual-containing cyclic codes over R. [ABSTRACT FROM AUTHOR]

We study algebraic geometry codes producing quantum error-correcting codes by the CSS construction. We pay particular attention to the family of Castle codes. We show that many of the examples known in the literature in fact belong to this family of codes. We systematize these constructions by showing the common theory that underlies all of them. [ABSTRACT FROM AUTHOR]

In this paper, we study quantum codes over $$F_q$$ from cyclic codes over $$F_q+uF_q+vF_q+uvF_q,$$ where $$u^2=u,~v^2=v,~uv=vu,~q=p^m$$ , and p is an odd prime. We give the structure of cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$ and obtain self-orthogonal codes over $$F_q$$ as Gray images of linear and cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$ . In particular, we decompose a cyclic code over $$F_q+uF_q+vF_q+uvF_q$$ into four cyclic codes over $$F_q$$ to determine the parameters of the corresponding quantum code. [ABSTRACT FROM AUTHOR]

In this paper, we present two new classes of binary quantum codes with minimum distance of at least three, by self-complementary self-dual orientable embeddings of 'voltage graphs' and 'Paley graphs in the Galois field $$GF(p^{r})$$ ', where $$p\in {\mathbb {P}}$$ and $$r\in {\mathbb {Z}}^{+}$$ . The parameters of two new classes of quantum codes are $$[[(2k'+2)(8k'+7),2(8k'^{2}+7k'),d_\mathrm{min}]]$$ and $$[[(2k'+2)(8k'+9),2(8k'^{2}+9k'+1),d_\mathrm{min}]]$$ , respectively, where $$d_\mathrm{min}\ge 3$$ . For these quantum codes, the code rate approaches 1 as $$k'$$ tends to infinity. [ABSTRACT FROM AUTHOR]

Quantum maximum-distance-separable (MDS) codes form an important class of quantum codes. It is very hard to construct quantum MDS codes with relatively large minimum distance. In this paper, based on classical constacyclic codes, we construct two classes of quantum MDS codes with parameters where $$2\le d\le (q+1)/2+\lambda -1$$ , and $$q+1=\lambda r$$ with $$r$$ even, and where $$2\le d\le (q+1)/2+\lambda /2-1$$ , and $$q+1=\lambda r$$ with $$r$$ odd. The quantum MDS codes exhibited here have parameters better than the ones available in the literature. [ABSTRACT FROM AUTHOR]

Two new families of asymmetric quantum codes are constructed in this paper. The first one is derived from the Calderbank-Shor-Steane (CSS) construction applied to classical Reed-Solomon (RS) codes, providing quantum codes with parameters [[ N = l( q−1), K = l( q−2 d + c + 1), d ≥ d/ d ≥ ( d− c)]], where q is a prime power and d > c + 1, c ≥ 1, l ≥ 1 are integers. The second family is derived from the CSS construction applied to classical generalized RS codes, generating quantum codes with parameters [[ N = mn, K = m(2 k− n + c), d ≥ d/ d ≥ ( d− c)]], where q is a prime power, 1 < k < n < 2 k + c ≤ q, k = n − d + 1, and n, d > c + 1, c ≥ 1, m ≥ 1 are integers. Although the second proposed construction generalizes the first one, the techniques developed in both constructions are slightly different. These new codes have parameters better than or comparable to the ones available in the literature. Additionally, the proposed codes can be utilized in quantum channels having great asymmetry, that is, quantum channels in which the probability of occurrence of phase-shift errors is large when compared to the probability of occurrence of qudit-flip errors. [ABSTRACT FROM AUTHOR]

We introduce simple qubit-encodings and logic gates which eliminate the need for certain difficult single-qubit operations in superconducting phase-qubits, while preserving universality. The simplest encoding uses two physical qubits per logical qubit. Two architectures for its implementation are proposed: one employing N physical qubits out of which N/2 are ancillas fixed in the |1 state, the other employing N/2+1 physical qubits, one of which is a bus qubit connected to all others. Details of a minimal set of universal encoded logic operations are given, together with recoupling schemes, that require nanosecond pulses. A generalization to codes with higher ratio of number of logical qubits per physical qubits is presented. Compatible decoherence and noise suppression strategies are also discussed. PACS: 03.67.Lx; 85.25.Hv; 03.67.-a; 89.70.+c [ABSTRACT FROM AUTHOR]