On Quantum and Classical BCH Codes 论文

摘要

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> 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 <emphasis><formula formulatype="inline"> <tex>$n$</tex></formula></emphasis> can contain its dual code only if its designed distance <emphasis><formula formulatype="inline"><tex>$\delta =O(\sqrt {n})$</tex></formula></emphasis>, 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. </para>