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

Abstract

A class of powerful q-ary linear polynomial codes originally proposed by Xing and Ling is deployed to construct good asymmetric quantum codes via the standard CSS construction. Our quantum codes are q-ary block codes that encode k qudits of quantum information into n qudits and correct up to (d x − 1)/2 bit-flip errors and up to (d z − 1)/2 phase-flip errors.. In many cases where the length (q 2 − q)/2 ≤ n ≤ (q 2 + q)/2 and the field size q are fixed and for chosen values of d x ∈ {2, 3, 4, 5} and d z ≥ δ , where δ is the designed distance of the Xing-Ling (XL) codes, the derived pure q-ary asymmetric quantum CSS codes possess the best possible size given the current state of the art knowledge on the best classical linear block codes.

Improved Constructions of Nested Code Pairs

“…, N s 2 } where the enumeration of the N i s is with respect (3) and (4) the right-hand sides of (21) and (22) therefore serve as lower bounds on their respective left-hand sides. We next prove (17) and (18). From (21) we have…”

“…The present example illustrates the huge advantage of using instead the relative minimum distances (which is what is behind the bounds in Corollary 29). This is done by investigating for each ℓ and δ what is the highest value g(ℓ, δ) such that the tables of best known linear codes in [34] guarantee the existence of linear code pairs A, B ⊥ satisfying dim A − dim B = ℓ, d(A) ≥ δ, and d(B ⊥ ) ≥ g(ℓ, δ) (this is in the spirit of [18,Theorem 2]). Observe, that we make no assumption whatsoever that B ⊂ A.…”

“…The literature only reports few families of long asymmetric quantum codes, an exception being La Guardia's construction II in [44,Theorem 7.1] which we shall compare our codes with. In another direction Ezerman et al's works in [17,18] explain how to derive good asymmetric quantum codes with d x being very small and ℓ being moderate. As our constructions do not seem to generally compare well with these codes in the case of d x ∈ {2, 3} we omit such cases in our tables.…”

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

“…Here, we recall some basic results of quantum convolutional code in [16,18,19,34,35]. The stabilizer can be given by a matrix of the form…”

“…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]. In [31], the author utilized some classes of cyclic codes to construct some good quantum convolutional codes compared with the ones in [2].…”