On the Construction of Optimal Asymmetric Quantum Codes

Quantum error-correcting codes have gained prominence since the initial discovery of Shor and Steane. In 1998, Calderbank et al. presented systematic methods to construct binary quantum codes, called stabilizer codes or additive codes, from classical error-correcting codes. Since then the field has made rapid progress, many good binary quantum codes were constructed by using classical error-correcting codes, such as BCH codes, Reed-Solomon codes, Reed-Muller codes, and algebraic geometric codes (see Refs. 4-8). The theory was later extended to the nonbinary case, since the realization that nonbinary quantum codes can use faulttolerant quantum computation (see Refs. 9-13). Recently, a number of new types of quantum codes, such as convolutional quantum codes, subsystem quantum codes have been studied and the stabilizer method has been extended to these variations of quantum codes (see Refs. 14-15). Asymmetric quantum error-correcting codes(AQECC) are quantum codes defined over quantum channels where qudit-flip errors and phase-shift errors may have different probabilities. AQECC was first studied by Steane in [16]. Since then, the construction of quantum codes have extended to asymmetric quantum channels. Loffe et al. utilize BCH codes to correct qubit-flip errors and LDPC codes to correct more frequently phase-shift errors. AQECC derived from LDPC codes and BCH codes were also constructed in [18-21]. Stephens et al. consider the investigation of AQECC via code conversion. Wang et al. presented the construction of nonadditive AQECC as well as constructions of asymptotically good AQECC derived from algebraic-geometry codes. Ezerman et al. presented the construction of AQECC under the trace Hermitian inner product. Ezerman and Ling studied two systematic construction of AQECC. Chee et al. constructed pure q-ary AQECC and some of these codes attain the quantum Singleton bound. Recently, Ezerman et al. also studied the pure AQECC and some optimal codes are obtained. A variety of the constructions of new AQECC were presented in [28-31]. AQECC attain the quantum Singleton bound are called optimal. Until now, just several families of optimal AQECC have been constructed. Chee et al. constructed optimal AQECC with parameters [[2 + 2, 2− 4, 4/4]]2m using generalized RS codes, where m is a positive integer. Guardia 28 constructed optimal AQECC with parameters [[p − 1, p − 2d + 2, d/(d − 1)]]p, where p is a prime number. Qian 32 constructed optimal AQECC with parameters [[q+1, q+1−2(k+i+2), (2k+3)/(2i+3)]]q2, where 0 ≤ k ≤ i ≤ q/2−1. Recently, Chen et al. constructed two families of optimal AQECC derived from negacyclic codes. In this paper, we constructed six new families of optimal AQECC derived from constacyclic codes. They are given by