@MISC{_(1)let, author = {}, title = {(1) Let Gi,Hi be the generator matrix and parity check matrix of Ci respectively, (i=1,2). Without loss of generality,}, year = {} }

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Abstract

between two distinct codewords, then C is called an [n,k,d] linear code over F 2. A linear code C is always specified by an n by k generator matrix G whose entries are all zeroes and ones. The generator matrix G maps k bits of information to a set of binary vectors of length n, called codewords. Binary linear codes can be alternatively (but equivalently) formulated by so called parity matrix, which is used to perform error-correction. The parity matrix H of a linear code [n,k] is an (n−k)×n matrix such that Hx=0 for all those and only those vectors x in the code C. The rows of H are n−k linearly independent vectors, and the code space is the space of vectors that are orthogonal to all of these vectors. A quantum error correcting code (QECC) Q:[[n,k,d]] is a 2 k-dimensional subspace of the Hilbert space C. It ⎡ d −1⎤ is a way of encoding k-qubit quantum states into n qubits (k<n) such that any error in ≤ ⎢ ⎥ qubits can be ⎣ 2 ⎦ measured and subsequently corrected without disturbing the encoded states. d is called the minimal distance of Q. Quantum CSS codes can be constructed by using classical linear codes. Theorem 1 [6] ⊥. Suppose that there exist two classical binary linear codes C1=[n,k1,d1],C2=[n,k2,d2], and ⊆C (so that n≤k 1+k 2). Then there exists a QECC 1 2 1 2 expressed as 2 n