6 Discussion This paper has studied how entanglement can be used to increase the minimum distance of quantum error-correcting codes. We demonstrated the encoding optimization procedure for EAQEC codes obtained by adding ebits to standard quantum stabilizer codes. The four types of unitary row operators play an important role in this encoding optimization procedure, and also help to clarify the properties of EAQEC codes and their relationship to standard codes. Some applications of the encoding optimization procedure were found to have promising results: we found [[7 , 1 , 5; 2]] and [[7 , 1 , 5; 3]] EAQEC codes from quantum BCH codes, and [[9 , 1 , 7; 4]] and [[9 , 1 , 7; 5]] EAQEC codes from Shor’s 9 − qubit code, together with a family of [[ n, 1 , n ; n − 1]] EAQEC codes for n odd, all of which achieve the quantum singleton bound. Several EAQEC codes found by this encoding optimization procedure are also degenerate codes. This procedure serves as an EAQEC code construction method for given parameters n, k, c . The encoding optimization procedure has very high complexity. However, it might be useful to further investigate it for specific families of codes that have special algebraic structures, such as quantum BCH codes and quantum Reed-Muller codes. For example, if we added ebits to a quantum Reed-Muller code (by the CSS construction with a classical RM ( r, m ) code [14]), we found that its simplified logical matrix L ′ is, like a check matrix of a CSS code, of the form    O L ′$_{Z}$ L ′$_{X}$ O    . By examining the symplectic relations, we found that the classical code, generated by L ′$_{Z}$ or L ′$_{X}$ , together with the RM ( r, m ) code, generate a subcode of RM ( r ′ , m ), which con- tains more lower-weight codewords. The minimum distance of the EAQEC code might be increased approximately by a factor 3 2 similar to the method used in [19] or [18]. From a [[16 , 6 , 4]] quantum Reed-Muller code, we obtained a [[16 , 6 , 6; 10]] EAQEC code by construct- 36