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Maximum distance separable (MDS) quantum error-correcting codes (QECCs) and MDS entanglement-assisted QECCs (EAQECCs) play significant roles in quantum information theory. In this paper, we construct several new families of MDS QECCs and MDS EAQECCs by utilizing Hermitian self-orthogonal generalized Reed–Solomon codes. These newly obtained MDS QECCs contain some known classes of MDS QECCs as subclasses and some of them have larger minimum distance. In addition, many q-ary MDS QECCs and MDS EAQECCs in our constructions have length exceeding q + 1 and minimum distance surpassing q 2 + 1 . [ABSTRACT FROM AUTHOR]

In this paper, we consider group rings and skew group rings and use them to construct binary quantum error-correcting codes(QECCs, for short). We study three classes of groups: direct product of two cyclic groups, dihedral group, direct product of cyclic group and dihedral group. And several special groups. Using these groups we can get construction matrix of the linear codes over F 4 . Then by computer search, we get 12 new binary QECCs with parameters [[40,16,7]], [[42,14,8]], [[48,20,8]], [[50,24,7]], [[54,34,6]], [[55,21,9]], [[60,38,6]], [[60,32,7]], [[70,42,7]], [[72,50,6]], [[75,47,7]], [[75,39,8]]. They both break the best-known lower bound on minimum distance in the Grassl’s code tables. [ABSTRACT FROM AUTHOR]

Quantum maximum-distance-separable (MDS) codes play an important role in the quantum codes. The previous quantum MDS codes had been constructed according to q is odd or even. However, such classification omits to consider some special categories of quantum MDS codes. Because of this, we will discuss the other classifications of q. In this paper, we construct some new q -ary quantum MDS codes from generalized Reed–Solomon codes by using Hermitian construction, and prove that these quantum MDS codes have minimum distance greater than q 2 , where q ≡ − 1 (mod 3). [ABSTRACT FROM AUTHOR]

In this paper, we use the principal ideals of group rings to construct quantum codes. We use multivariate polynomial rings to represent group rings and use multivariate polynomials to represent codewords. We give a condition for multivariate polynomials such that they generate Hermitian dual-containing codes. We also find that the minimum distance of the code which is generated by a multivariate polynomial is related to the zeros distribution of the multivariate polynomial. After computer search and calculation, we get many quantum codes with good parameters over small finite fields. [ABSTRACT FROM AUTHOR]

Quantum convolutional codes, which are the correct generalization to quantum domain of their classical analogs, were introduced to overcome decoherence during long distance quantum communications. In this paper, we construct some classes of quantum convolutional codes via classical constacyclic codes. These codes are maximum-distance-separable (MDS) codes in the sense that they achieve the Singleton bound for pure convolutional stabilizer codes. Furthermore, compared with some of the codes available in the literature, our codes have better parameters and are more general. [ABSTRACT FROM AUTHOR]

Hermitian dual-containing codes play an important role in constructing quantum codes, constructing Hermitian dual-containing codes with optimal parameters is an interesting topic. The objective of this paper is to construct several classes of quantum codes with optimal parameters. Using cyclic codes, three classes of optimal Hermitian dual-containing codes are constructed. By employing the Hermitian construction, three classes of quantum codes with optimal parameters are obtained. These codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

Quantum synchronizable codes(QSCs) are special classes of quantum error-correcting codes that can correct not only the effects of quantum noise on qubits but also the misalignment in block synchronization. In this paper, we construct two classes of QSCs from cyclic codes. The first class is to construct cyclic and negacyclic codes of odd prime length by using cyclotomic polynomials, and obtain a class of QSCs which achieves the best attainable misalignment under certain conditions. The second class is to apply BCH codes and negacyclic BCH codes on (u + v)|(u − v) structure to construct a class of QSCs. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes, which is a generalization of quantum error-correcting codes, could be derived from any classical codes by utilizing pre-shared entangled states between the sender and the receiver. In this paper, we construct some entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes from constacyclic codes and cyclic codes by exploiting less pre-shared entangled states, respectively. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. In particular, we extend the results in Tian et al. (Int. J. Theor. Phys. 60, 1843–1857, 2021) to more general case. [ABSTRACT FROM AUTHOR]

In this work, we propose two criteria for linear codes obtained from the Plotkin sum construction being symplectic self-orthogonal (SO) and linear complementary dual (LCD). As specific constructions, several classes of symplectic SO codes with good parameters including symplectic maximum distance separable codes are derived via ℓ -intersection pairs of linear codes and generalized Reed-Muller codes. Also symplectic LCD codes are constructed from general linear codes. Furthermore, we obtain some binary symplectic LCD codes, which are equivalent to quaternary trace Hermitian additive complementary dual codes that outperform the best-known quaternary Hermitian LCD codes reported in the literature. In addition, we prove that symplectic SO and LCD codes obtained in these ways are asymptotically good. [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]

Galois hulls are generalizations of Euclidean and Hermitian hulls, which have attracted interest because of their important applications in determining the complexity of some algorithms about linear codes and in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, the whole contribution is two-folded. On one hand, we propose explicit methods to construct linear codes of larger lengths with Galois hulls of arbitrary dimensions from given Galois self-orthogonal codes. Conditions required in our approach are proven to be relatively weak. On the other hand, we apply these results to construct EAQECCs. Two bounds for EAQECCs constructed from linear codes with prescribed dimensional Galois hull are given and EAQECCs with rates greater than or equal to 1 2 and positive net rates can be obtained. We also present many interesting examples to explain visually how these two aspects work. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum maximum distance separable (MDS) codes form a significant class of quantum codes. By using constacyclic codes, we construct some new classes of q-ary entanglement-assisted quantum error-correcting MDS codes. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of q -ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths n | (q 2 − 1) with flexible parameters as to the minimum distance d and the number c of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature. [ABSTRACT FROM AUTHOR]

Let R = 𝔽 q + u 𝔽 q , where q is a power of a prime number p and u 2 = 0. A triple cyclic code of length (r , s , t) over R is a set that can be partitioned into three parts that any cyclic shift of the coordinates of the three parts leaves the code invariant. These codes can be viewed as R [ x ] -submodules of R [ x ] / 〈 x r − 1 〉 × R [ x ] / 〈 x s − 1 〉 × R [ x ] / 〈 x t − 1 〉. In this paper, we study the generator polynomials and the minimum generating sets of this kind of codes. Some optimal or almost optimal linear codes are obtained from this family of codes. We present the relationship between the generators of triple cyclic codes and their duals. As a special class of triple cyclic codes, separable codes over R are discussed briefly in the end. [ABSTRACT FROM AUTHOR]

Recently, the research of trace codes over finite fields has received a lot of attention because some of these codes have good parameters. In this paper, we construct several classes of trace codes from some special classes of linear codes and give their weight distributions, showing that they contain some optimal and almost optimal codes. Furthermore, we determine the parameters of the duals of the trace codes, and show that they are distance-optimal or nearly distance-optimal with respect to the sphere-packing bound. [ABSTRACT FROM AUTHOR]

Classical Bose-Chaudhuri-Hocquenghem (BCH) codes over finite fields have studied extensively. The quantum stabilizer codes with good parameters can be constructed using classical BCH codes. In this paper, we study the construction of quantum codes from nonprimitive BCH codes over the finite field F q 2 . Taking advantage of Hermitian dual containing BCH codes of length n = r(q2 − 1) over F q 2 where ordn(q2) = 4, we can construct some families of new quantum codes with good parameters. [ABSTRACT FROM AUTHOR]

In this paper, we construct four families of entanglement-assisted quantum error-correcting codes (EAQECCs) with flexible parameters from negacyclic codes. By selecting different defining sets, our codes are new in the sense that their parameters are not covered by the codes available in the literature. Moreover, the EAQECCs we constructed are optimal and have larger minimum distance compared with the known codes with the same length. [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]

The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes. In this work, we construct some new entanglement-assisted quantum maximum distance separable codes with length n = q 2 + 1 5 from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum maximum distance separable (MDS) codes form a significant class of quantum codes. By using generalized Reed–Solomon (GRS) codes and extended GRS codes, we construct some new classes of q-ary entanglement-assisted quantum error-correcting MDS codes. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we construct some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes with lengths n = q 2 + 1 and n = (q 2 + 1) / 2 from negacyclic MDS codes and constacyclic MDS codes, respectively. All of them have flexible parameters. These EAQMDS codes we constructed have larger minimum distance and contain the known EAQMDS codes with same length in previous papers. [ABSTRACT FROM AUTHOR]

In this paper, we construct optimal or almost optimal dual-containing cyclic codes from cyclotomic classes of order r. Based on these cyclic codes constructed, we obtain many new quantum codes comparing with the known literatures. Furthermore, we construct some quantum synchronizable codes with good error-correcting ability towards bit errors and phase errors by a pair of cyclic codes with special containing property. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes are a generalization of standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing pre-shared entanglement between the sender and the receiver. In this paper, we construct seven new families of entanglement-assisted quantum maximum-distance-separable codes from cyclic codes by exploiting less pre-shared entangled states. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature. [ABSTRACT FROM AUTHOR]

BCH codes are an important class of cyclic codes which have applications in satellite communications, DVDs, disk drives, and two-dimensional bar codes. Although BCH codes have been widely studied, their parameters are known for only a few special classes. Recently, Ding et al. made some new progress in BCH codes. However, we still have very limited knowledge on the dimension of BCH codes, not to mention the weight distribution of BCH codes. In this paper, we generalize the results on BCH codes from several previous papers. 1) The dimension of narrow-sense BCH codes of length $((q^{m}-1)/{\lambda })$ with designed distance $2\leq \delta \leq (({q^{\lceil (m+1)/2 \rceil }-1})/(\lambda)+1)$ is settled, where $\lambda $ is any factor of $(q-1)$. 2) The weight distributions of two classes of narrow-sense BCH codes of length $(({q^{m}-1})/2)$ with designed distance $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m-1)/2\rfloor }-1})/2)$ and $\delta =(({(q-1)q^{m-1}-q^{\lfloor (m+1)/2\rfloor }-1})/2)$ are determined. 3) The weight distribution of a class of BCH codes of length $((q^{m}-1)/({q-1}))$ is determined. In particular, a subclass of this class of BCH codes is optimal with respect to the Griesmer bound. Some optimal linear codes obtained from this class of BCH codes are characterized. [ABSTRACT FROM AUTHOR]

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed–Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of bits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of q 2 - 1 , which are completely new and unlike all those known lengths existed before. [ABSTRACT FROM AUTHOR]

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field F q has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes. [ABSTRACT FROM AUTHOR]

The entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and important class of quantum codes. The entanglement-assisted formalism can transform arbitrary classical linear codes into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, by decomposing the defining set of negacyclic BCH codes, we construct a class of new EAQECCs with length n = q 4 m − 1 q 2 − 1 . [ABSTRACT FROM AUTHOR]

In this paper, we give the structures of all ( 1 + u + v ) -constacyclic codes of length 2 s over F 2 + u F 2 + v F 2 + u v F 2 , we also obtain the number of codewords, and the dual of each constacyclic code. Among other results, we study the form of such constacyclic codes that are self-dual and give the number of these codes. We end by giving all self-dual constacyclic codes of length 2, 4 and 8. [ABSTRACT FROM AUTHOR]

Dynamic quantum secret sharing occupies an important position in quantum cryptography. In this paper, an efficient and secure dynamic quantum direct two-secrets sharing scheme is proposed based on the GHZ state. The proposed scheme is a one-time sharing of a determined classic message and quantum states. For recovering the secret messages, the agents only need to have the ability to perform X-basis measurements on the particles without performing any unitary operation. When dynamically deleting agents, our scheme only requires the other agents to announce some information without transmitting any quantum. Moreover, the analysis shows that our scheme is able to resist dishonest revoked agent attack as well as a range of other common attacks. Compared with the existing dynamic quantum direct secret sharing schemes, the proposed scheme is more efficient and more secure. [ABSTRACT FROM AUTHOR]

Hermitian dual-containing codes are an important class of linear codes which have important applications in the construction of quantum codes. In this paper, two classes of Hermitian dual-containing almost MDS codes over finite fields are studied. By employing the Hermitian construction, a class of quantum codes with minimum distance 3 and a class of quantum codes with minimum distance 4 are constructed. [ABSTRACT FROM AUTHOR]

Let q be an even prime power and ω be a primitive element of 픽q2. By analyzing the structure of cyclotomic cosets, we determine a sufficient condition for ωq− 1-constacyclic codes over 픽q2 to be Hermitian dual-containing codes. By the CSS construction, two classes of new optimal AQECCs are obtained according to the Singleton bound for AQECCs. [ABSTRACT FROM AUTHOR]

Motivated by the works of Shiromoto [3] and Shi et al. [4] , we study the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over the ring Z ℓ . Necessary and sufficient conditions for the existence of the MacWilliams type identities with respect to the Lee and Euclidean weight enumerators for linear codes over Z ℓ are given. Some examples about such MacWilliams type identities are also presented. [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]

In distributed storage systems, locally repairable codes (LRCs) can effectively reduce the complexity of repairing failed nodes. In this paper, we constructed six classes of optimal cyclic (r , δ) -LRCs with unbounded length and minimum distance δ + 2 ≤ d ≤ 2 δ from their zeros. Two classes of optimal cyclic (r , δ) -LRCs with large minimum distance are also presented. [ABSTRACT FROM AUTHOR]

Recently, symbol-pair codes are proposed to protect against pair errors in symbol-pair read channels. One main task in symbol-pair coding theory is to design codes with large minimum pair distance. Maximum distance separable (MDS) symbol-pair codes are optimal in the sense they attain maximal minimum pair distance. In this paper, based on constacyclic codes, we construct some new MDS symbol-pair codes with minimum pair-distance five and six. Compared with classical $q$ -ary MDS codes, the constructed MDS symbol-pair codes have length up to q^{2}+q+1 . [ABSTRACT FROM PUBLISHER]

Quantum secret sharing is an important technique of quantum cryptography. A (t, n) threshold quantum secret sharing scheme with fairness is proposed. Firstly, the proposed scheme only requires the distributor to provide a share for each participant to achieve fairness. Secondly, the distributor interacts with participants only during the secret distribution phase. Another important point is that our scheme combines the privacy features of secure multi-party computing to ensure the reuse of participants’ secret shares. Finally, in our scheme, the correctness, security and fairness are discussed in detail. [ABSTRACT FROM AUTHOR]

Quantum secret sharing is an important technology in quantum cryptography, and it is used in the fields of secure multiparty computation, quantum auction and fiber network configuration. In this paper, we construct two quantum secret sharing schemes with adversary structure by using three-qudit GHZ state and three qubits. In the proposed scheme, the dealer first generates the private keys based on the adversary structure, then the participants in authorized subset use their private keys to perform the unitary operations on the quantum state and recover the original secret. The security analysis shows the attacks of intercept-and-resend, entangle-and-measure and participant are impossible in our schemes. Compared with the existing quantum secret sharing schemes, we realize the sharing of classical information and quantum state, and need less computational cost. Moreover, our schemes are more efficient when the adversary structure can be obtained directly. [ABSTRACT FROM AUTHOR]

In this paper, we study cyclic self-orthogonal codes over Z 2 m for any odd length. We give the generator polynomials of cyclic self-orthogonal codes over Z 2 m . By using these generator polynomials, we obtain a sufficient and necessary condition for the existence of nontrivial cyclic self-orthogonal codes over Z 2 m and determine the number of such codes for a given odd length. Some results about cyclic self-dual codes over Z 2 m are included. [ABSTRACT FROM AUTHOR]

One of the most challenges to prove the feasibility of quantum computers is to protect the quantum nature of information. Quantum convolutional codes are aimed at protecting a stream of quantum information in a long distance communication, which are the correct generalization to the quantum domain of their classical analogs. In this paper, we construct some classes of quantum convolutional codes by employing classical constacyclic codes. These codes are optimal in the sense that they attain the Singleton bound for pure convolutional stabilizer codes. [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]

The hull of a linear code is the intersection of itself with its dual code with respect to certain inner product. Both Euclidean and Hermitian hulls are of theorical and practical significance. In this paper, we construct several new classes of maximum distance separable (MDS) codes via (extended) generalized Reed-Solomon (GRS) codes and determine their Euclidean or Hermitian hulls. As a consequence, four new classes of MDS codes with Hermitian hulls of flexible dimensions and six new classes of MDS codes with Euclidean hulls of flexible dimensions are constructed. As applications, for the former, we further construct four new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and four new families of MDS EAQECCs of length n > q + 1 . Meanwhile, many of the distance parameters of our MDS EAQECCs are greater than ⌈ q 2 ⌉ or q; for the latter, we show some examples on Euclidean self-orthogonal and one-dimensional Euclidean hull MDS codes. In addition, two new general methods for constructing extended GRS codes with (k - 1) -dimensional Hermitian hull and Hermitian self-orthogonal extended GRS codes are also provided. [ABSTRACT FROM AUTHOR]

Quantum secure multiparty summation plays an important role in quantum cryptography. In the existing quantum secure multiparty summation protocols, the (n,n)-threshold protocol has been given extensive attention. To increase the applicability of quantum secure multiparty summation protocols, a new quantum secure multiparty summation protocol based on Shamir’s threshold scheme and d-dimensional GHZ state is proposed in this paper. In the proposed protocol, i) it has a (k,n)-threshold approach; ii) in the result output phase, it can not only detect the existence of deceptive behavior but also determine the specific cheaters; iii) compared with the (n,n)-threshold quantum secure multiparty summation protocols, it needs less computation cost when L satisfies L > 4, where L is the length of each participant’s secret. In addition, the security analysis shows that our protocol can resist intercept-resend attack, entangle-measure attack, Trojan horse attack, and participant attack. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, based on the Steane's enlargement construction, three classes of non-binary quantum codes are constructed from classical repeated-root cyclic codes of length 2ps over 픽q with odd characteristic p. The exact minimum distances of these quantum codes are determined. This construction yields a quantum MDS code with parameters [[2p, 2p - 4, 3]]p and two good quantum codes with parameters [[2p, 2p - 7, 4]]p and [[2p, 2p - 10, 5]]p. [ABSTRACT FROM AUTHOR]

One central theme in quantum error-correction is to construct quantum codes that have a large minimum distance. Quantum maximal distance separable (MDS) codes are optimal in the sense they attain maximal minimum distance. Recently, constructing quantum MDS codes has received much attention and seems to become more and more difficult. In this paper, based on classical constacyclic codes, we construct some new quantum MDS codes by employing the Hermitian construction. Compared with the known quantum MDS codes, these quantum MDS codes have much larger minimum distance. [ABSTRACT FROM PUBLISHER]

The error linear complexity spectrum of a periodic sequence is defined to be the ordered list of $$k$$ k -error linear complexities of the sequence. In this paper, we present an algorithm which computes the error linear complexity spectrum for binary sequences with period $$p^{n}$$ p n , where $$p$$ p is an odd prime and $$2$$ 2 is a primitive root modulo $$p^{2}$$ p 2 . [ABSTRACT FROM AUTHOR]

Let q be a power of an odd prime, and be the trace mapping. A mapping is called planar (or perfect nonlinear) on if for any non-zero , the difference mapping is a permutation where for , . Kyureghyan and Özbudak (2012) [8] considered the planarity of mappings on for and proved that there is no planar for . For the case and , they raised three conjectures. In this paper we prove the third conjecture which says that there is no planar for , by using Kloosterman sums. Our proof also works for case , so we present a new proof of the Kyureghyan–Özbudak result. For case , we present an elementary proof of the first conjecture which says that there is no planar for . [ABSTRACT FROM AUTHOR]

Let q be an odd prime power. Based on classical negacyclic codes, we construct two classes of quantum maximum-distance-separable (MDS) codes with parameters $ [[q^2+1, q^2-2d+3, d]]q $where q\equiv 1\;(\bmod\;4) and 2 \leq d \leq q+1 is even, and $ [[(q^2+1)/2,(q^2+1)/2-2d+2,d]]q $where 3 \leq d \leq q is odd. Some of these quantum MDS codes are new in the sense that their parameters are different from all the previously known ones. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we investigate negacyclic self-dual codes over using the higher torsion codes. It is proved that all negacyclic self-dual codes over are of Type I. Examples of extreme Type I codes over from negacyclic codes are given. [Copyright &y& Elsevier]

As one of the primitive operations of quantum cryptography, quantum secure multiparty multiplication plays an important role in practical applications. In this paper, we propose a (k, n)-threshold dynamic quantum secure multiparty multiplication protocol. (i) Based on Shamir's threshold scheme, the threshold (k, n) can be implemented; (ii) In the multiparty multiplication phase, the cheating behavior of participants can be detected by the one-to-one correspondence of hash values; (iii) Any m participants can dynamically join or exit during the execution of the protocol. Moreover, the method of this paper can also achieve dynamic quantum secure multiparty summation. Further, the security analysis shows that our protocol is resistant to intercept-resend attack, entangle-measure attack, Trojan horse attack, and participant attack. [ABSTRACT FROM AUTHOR]