Asymmetric quantum codes: new codes from old

Abstract

In this paper we extend to asymmetric quantum error-correcting codes (AQECC) the construction methods, namely: puncturing, extending, expanding, direct sum and the (u|u + v) construction. By applying these methods, several families of asymmetric quantum codes can be constructed. Consequently, as an example of application of quantum code expansion developed here, new families of asymmetric quantum codes derived from generalized Reed-Muller (GRM) codes, quadratic residue (QR), Bose-Chaudhuri-Hocquenghem (BCH), character codes and affine-invariant codes are constructed.

Some families of asymmetric quantum codes and quantum convolutional codes from constacyclic codes

“…In this section, we state some definitions and some basic results in [28][29][30]32,33] firstly, then we will use constacyclic codes from [11] to construct some families of optimal asymmetric quantum codes.…”

“…In many quantum mechanical systems, the occurrence of qubit-flip and phase-shift errors is quite different [12]. For the past two decades, the constructions of good asymmetric quantum codes have been investigated by some researchers [28][29][30]32,33,38]. Qian and Zhang utilized q 2 -ary cyclotomic cosets to construct a family of optimal asymmetric quantum codes in [43].…”

Asymmetric Entanglement-Assisted Quantum Error-Correcting Codes and BCH Codes

“…In [27], the authors noticed that phaseshift errors happened more likely than qudit-flip errors, thus it was desirable to construct quantum codes where two minimum distances d x and d z , for detecting qudit-flip and phaseshift errors, respectively, were considered and provide results for addressing their behavior. As a consequence, in the last years asymmetric quantum error-correcting codes have been studied giving rise to codes suitable when dephasing occurs more often than relaxation [14,15,16,31,32,40]. Most of the asymmetric quantum codes come from the CSS construction of quantum stabilizer codes and, for them, there is also a Gilbert-Varshamov bound [35].…”

On optimal constacyclic codes

“…In the last two decades, many researches have focussed the attention in the investigation of properties of asymmetric quantum error-correcting codes (AQECC) [7,18,23,30,32,33], as well as constructions of AQECC with good or optimal parameters (optimal in the sense that the code parameters attain the quantum version of Singleton bound [30,Lemma 3.2] with equality; these codes are called maximum-distance-separable or MDS).…”