On Quantum and Classical BCH Codes
Salah A. Aly, Andreas Klappenecker, Pradeep Kiran Sarvepalli · IEEE Transactions on Information Theory · 2007
Classical Bose–Chaudhuri–Hocquenghem (BCH) codes that contain their (Euclidean or Hermitian) dual codes can be used to construct quantum stabilizer codes; this correspondence studies the properties of such codes. It is shown that a BCH code of length$n$can contain its dual code only if its designed distance$\delta =O(\sqrt {n})$, and the converse is proved in the case of narrow-sense codes. Furthermore, the dimension of narrow-sense BCH codes with small design distance is completely determined, and – consequently – the bounds on their minimum distance are improved. These results make it possible to determine the parameters of quantum BCH codes in terms of their design parameters.