Asymmetric quantum Griesmer codes detecting a single bit-flip error

The quest to build large-scale quantum computing devices depends on keeping the noise level below a fault-tolerance threshold. In this paper we derive the asymmetric quantum analogue of the Griesmer bound. To benefit from the noise asymmetry in many physical systems, one can decide to only detect a single bit-flip error while maximizing control over the phase-flip errors. We present constructions of such codes via the classical Griesmer codes and obtain infinite families. The optimality of the parameters of the codes in the families is measured against the quantum Griesmer bound. Numerous other codes, which may not be optimal, can also be derived. Choices of their design provide greater flexibility in terms of the resulting quantum parameters. We give examples of good qubit, qutrit, and ququad codes from such a route.