On Quantum Hamming Bound

It is desirable to study upper and lower bounds on the minimum distance and dimensions of quantum codes, so the computer search on the code parameter can be minimized and optimal codes can be known. It is a well-known fact that Singleton and Hamming bounds hold for classical codes [9]. We need some bounds on the achievable minimum distance of a quantum stabilizer code. Perhaps the simplest one is the Knill-LaFlamme bound, also called the quantum Singleton 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].