Pure Asymmetric Quantum MDS Codes From CSS Construction: A Complete Characterization

Abstract

Using the Calderbank-Shor-Steane (CSS) construction, pure q-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

Xing–Ling codes, duals of their subcodes, and good asymmetric quantum codes

“…Remark 4 For (q,t) = (4, 0) the resulting AQCs have either inferior parameters to or the same parameters as those of AQMDS codes already discussed in [4]. For q = 4, 1 ≤ t ≤ 4, there exists an AQC with parameters…”

“…In all other instances, d z = δ . We exclude CSS asymmetric quantum MDS codes from our tables as their treatment is already available in [4].…”

“…The tables provided in [8] and the references therein provide a summary of previously constructed families of AQCs. Note that most of the results in the MDS family presented in [8, Table 2] were mistakenly attributed to [12] instead of to the correct source [4]. This paper derives q-ary CSS AQCs with good parameters based on nested polynomial codes first introduced by Xing and Ling in [11] and, henceforth, called XL codes.…”

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

“…In [12], we studied the optimal asymmetric quantum codes by using negacyclic codes. For more constructions of asymmetric quantum codes, the readers can consult [13][14][15]37] for more detail. Now, the constructions of good quantum convolutional codes have been studied by many authors [1-3, [16][17][18][19]41,42].…”

On the construction of optimal asymmetric quantum codes

“…Recently, Ezerman et al 27 also studied the pure AQECC and some optimal codes are obtained. A variety of the constructions of new AQECC were presented in [28][29][30][31].…”