Construction of Binary Quantum Error-Correcting Codes from Orthogonal Array

Abstract

By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.

On generating r-uniform subspaces with the isometric mapping method

“…One example in this direction could be analyzing encoding isometries of the QECCs obtained with OAs, for instance, the ones from Ref. [25]. This could potentially lead to new OA and MOA constructions.…”

“…In the present paper we concentrate on construction of r-uniform subspaces, mostly for heterogeneous systems, i. e., those having different local dimensions. There are a number of tools for constructing r-uniform states in homogeneous systems: graph states [19], elements of combinatorial design such as Latin squares [20], symmetric matrices [21], orthogonal arrays (OAs) [22] and their variations [23][24][25]. For construction of r-uniform states in heterogeneous systems OAs were extended to mixed orthogonal arrays (MOAs) [26].…”

Construction of quaternary quantum error-correcting codes via orthogonal arrays

“…., (q Proof: An IrOA (4 8 , 16, 4, 5) with MD = 6 obtained by using product of two OA (2 8 , 16, 2, 5)s in [49] and an IrOA (4 12 , 24, 4, 7) with MD = 8 obtained by using product of two OA (2 12 , 24, 2, 7)s in [49] can generate two new QECCs ((16,1,6)) 4 and ((24,1,8)) 4 respectively. By using product of two OA (4608,23,2,4)s obtained from the ((23,9,5)) QECC in Example 7 in [15], we can get an OA (4608 2 , 23, 4, 4) (20,256,4)) 4 respectively. In particular, an IrOA (64,6,4,3) in [48] yields an optimal QECC ((6,1,4)) 4 in [50].…”

“…They play an important role in quantum information tasks, such as in entanglement purification, quantum key distribution, fault-tolerant quantum computation, and so on [6][7][8]. Since its discovery, code construction has come a long way [9][10][11][12][13][14][15][16]. Plenty of binary QECCs have been obtained, some of which from classical error-correcting codes (CECCs) [16][17][18][19].…”

On the Construction of Variable Strength Orthogonal Arrays

“…The construction of OAs has been made new progress recently. New construction methods are constantly proposed in Pang et al [29], Wang et al [30], Pang et al [31], Pang et al [32], Zhang et al [33], Pang et al [34], Du et al [35], Pang et al [36] and Pang et al [37], which could facilitate the construction of related structures to OAs. Moreover, communications and computer sciences often benefit from OAs and related structures.…”