Entanglement-Assisted and Subsystem Quantum Codes: New Propagation Rules and Constructions

Abstract

This paper proposes new propagation rules on quantum codes in the entanglement-assisted and in quantum subsystem scenarios. The rules lead to new families of such quantum codes whose parameters are demonstrably optimal. To obtain the results, we devise tools to puncture and shorten codes in ways that ensure their Hermitian hulls have certain desirable properties. More specifically, we give a general framework to construct k-dimensional generalized Reed-Solomon codes whose Hermitian hulls are (k − 1)-dimensional maximum distance separable codes.

Steane enlargement of Entanglement-Assisted Quantum Error-Correcting Codes

“…For the remaining cases, we have no generic candidate for A 0 such that rank(A 0 − A T 0 ) = t ℓ − 2, however for specific cases and moderate values of q and t ℓ , it is not hard to find suitable polynomials η t ℓ and attached matrices A 0 with the above mentioned rank. Note also that the propagation rules stated in [35] does not work here since our codes do not come from the Hermitian construction.…”

The hull of two classical propagation rules and their applications

“…Notably, in the whole process of development of coding theory, these propagation rules are widely used in quantum stabilizer codes, entanglement-assisted quantum error-correcting codes, subsystem codes, locally repairable codes, constant-weight codes, linear complement dual (LCD) codes, etc. (e.g., see [2,13,18,19,26,27,28,31,32] and the references therein) and many remarkable results were obtained. It is worth noting that recently the (u, u + v)-construction had also been used to design a novel Niederreiter-Like cryptosystem [39].…”

New MDS Entanglement-Assisted Quantum Codes from MDS Hermitian Self-Orthogonal Codes

“…In the above construction of EAQEC codes, the dimension h of the Euclidean or Hermitian hull of a linear code is a key parameter to control the dimension and consumption parameter of an EAQEC code. This is an important motivation to construct equivalent linear codes with various hull dimensions, see [15,16,19,22,[33][34][35]. An EAQEC code attaining this quantum Singleton bound is called an MDS EAQEC code.…”

“…An EAQEC code attaining this quantum Singleton bound is called an MDS EAQEC code. The construction of MDS QAECC code with large ranges of four parameters has been addressed in [15,16,19,22,[33][34][35] and references therein.…”