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One of the most important functions used in a quantum key distribution (QKD) network is universal hash functions, specially, (almost) strongly universal hash functions which are used in at least three steps of QKD, in particular, in error correction, privacy amplification, and authentication. Also, they have been recently used in several other quantum communication protocols like quantum secret sharing (QSS). These hash functions have also many other important applications from information security to data structures and parallel computing. Recently, Bibak et al. [Quantum Inf. Comput., 2021] introduced quadratic hash which gives much better collision bound than the well-known polynomial hash. In this paper, we define three new universal hash function families which strongly generalize all the previous families and have several advantages over them. Then, using highly influential and pioneering results of Schmidt and of Weil, we show that these new families are (almost) Δ -universal which can then be easily converted to (almost) strongly universal families. This makes them useful for applications in QKD and many other areas. [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]

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]

In quantum key distribution (QKD), two users extract a shared secret key using a quantum communication channel in the presence of an eavesdropper. Among QKD protocols, the ones based on energy-time (ET) entanglement of photons have been studied extensively due to their ability to generate high key rates from the arrival times of entangled photons. For the information reconciliation (IR) stage of ET-QKD protocols (where the users communicate using a classical channel in order to reconcile differences in their data), a scheme called multi-level coding (MLC) was proposed by Zhou et al. in prior work. The MLC scheme splits the raw key symbols into bit layers and utilizes binary low-density parity check (LDPC) codes to encode each layer. Although binary LDPC codes are able to offer low complexity decoding for IR, they have poor error-correcting performance compared to their non-binary counterparts, thus leading to low key rates. Additionally, existing LDPC codes do not fully utilize the properties of the QKD channel to optimize the key rates. In this paper, we mitigate the above issues by proposing a flexible protocol for IR in ET-QKD systems called non-binary multi-level coding NB-MLC(a) which is parameterized by a positive integer a. The NB-MLC(a) protocol is a generalization of the MLC scheme and utilizes NB-LDPC codes from a Galois field of size 2 a . We show that by using a small value of a, the NB-MLC(a) protocol significantly improves the key rate without much increase in complexity. To further improve the key rates of the NB-MLC(a) protocol, we propose (i) a joint rate and degree distribution optimization (JRDO) algorithm to design the NB-LDPC codes for the protocol and (ii) an interleaved decoding and communication (IDC) scheme to decode the different layers of the NB-MLC(a) protocol. The JRDO algorithm is designed to use the QKD channel information, and we show that it results in a higher key rate than codes used in prior work. Additionally, the IDC scheme improves the key rate compared to the decoding and communication methods utilized previously in literature. Overall, the NB-MLC(a) protocol that uses JRDO-LDPC codes and the IDC scheme results in a significant 40–60 % improvement in key rates compared to prior work for ET-QKD systems. [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 this paper, we provide methods for constructing Hermitian dual-containing (HDC) matrix-product codes over F q 2 from some non-singular matrices and a special sequence of HDC codes and determine parameters of obtained matrix-product codes when the input matrix and sequence of HDC codes satisfy some conditions. Then, using some nested HDC BCH codes with lengths n = q 4 - 1 a (a = 1 or a = q ± 1) , we construct some HDC matrix-product codes with lengths N = 2n or 3n and derive nonbinary quantum codes with length N from these matrix-product codes via Hermitian construction. Four classes of quantum codes over F q ( 3 ≤ q ≤ 5 ) are presented, whose parameters are better than those in the literature. Besides, some of our new quantum codes can exceed the quantum Gilbert-Varshamov (GV) bound. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting (EAQEC) codes could generalize and improve performance of standard quantum error-correcting (QEC) codes to a great extent. In this paper, series of EAQEC codes of length n = q - 1 a (q + 1) are constructed from cyclic codes and negacyclic codes, where q is a prime power and a is a positive integer such that a ∣ (q - 1) . It turns out that the number of required entanglement bits can take almost all possible values. Consequently, our EAQEC codes have flexible parameters and most of them are new. For given the same length, our construction contain and extend those known consequences in Grassl et al. (Int J Quantum Inf 2(1):55–64, 2004), Jin et al. (IEEE Trans Inf Theory 56:4735–4740, 2010), Kai et al. (IEEE Trans Inf Theory 60:2080–2086, 2014), Jin and Xing (IEEE Trans Inf Theory 60:2921–2925, 2014), Chen et al. (IEEE Trans Inf Theory 61:1474–1484, 2015), Zhang and Ge (IEEE Trans Inf Theory 61:5224–5228, 2015; Des Codes Cryptogr 83(3):503–517, 2016), Shi et al. (Cryptogr Commun 10(6):1165–1182, 2018; Finite Fields Appl 46:347–362, 2017), Fan et al. (Quantum Inf Comput 16:423–434, 2016), Lu et al. (Quantum Inf Process 17(69):1–23, 2018), Li et al. (Int J Quantum Inf 17(1):1950022, 2019), Liu et al. (Quantum Inf Process 17(210):1–19, 2018), Fang et al. (Euclidean and Hermitian Hulls of MDS Codes and Their Applications to EAQECCs. arXiv:https://arxiv.org/abs/1812.09019v3). Above all, all our codes are maximum-distance-separable (MDS) if their minimum distance d ≤ n + 2 2 . [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]