A Note on Quantum Hamming Bound

Quantum stabilizer codes are a known class of quantum codes that can protect quantum information against noise and decoherence. Stabilizer codes can be constructed from self-orthogonal or dualcontaining classical codes, see for example [3, 8, 11] and references therein. It is desirable to study upper and lower bounds on the minimum distance of classical and quantum codes, so the computer search on the code parameters can be minimized. It is a well-known fact that Singleton and Hamming bounds hold for classical codes [10]. Also, upper and lower bounds on the achievable minimum distance of quantum stabilizer codes are needed. Perhaps the simplest upper bound is the quantum Singleton bound, also known as the Knill-Laflamme bound. The binary version of the quantum Singleton bound was first proved by Knill and Laflamme in [12], see also [1, 2], and later generalized by Rains using weight enumerators in [16].