Applying the encoding circuit algorithm to this check matrix, we obtain an encoding operator U$_{E}$ . In particular, if C = O in (18), then When k = 1 , AB T + BA T = O holds if and only if A = B or at least one of A and B is the zero vector. Let W be the selection operator determined by the encoding circuit algorithm with A = [ 0 0 0 0 ] T and B = [ 1 0 1 0 ] T . Then the encoding operator U$_{E}$W generates a [[5 , 1 , 5; 4]] EAQEC code with a simplified check matrix AB T + BA T = O.        00000 11000 00000 01100 00000 00110 00000 00011 01111 00000 11000 00000 00011 00000 11110 00000           11111 00000 00000 11111    . and a simplified logical matrix With the help of 4 ebits, the minimum distance is increased from 1 to 5. The quantum singleton bound (11) is saturated by the parameters [[5 , 1 , 5; 4]]. Because the minimum distance of a standard [[9 , 1]] quantum stabilizer code is at most 3, this [[5 , 1 , 5; 4]] code is not equivalent to any standard 9-qubit code. ✷ In Example 1, we constructed a [[5 , 1 , 5; 4]] EAQEC code from a [[5 , 1 , 1]] 5-qubit bit flip 17