Construction Of [[n,n-6-4]]_q Quantum MDS Codes And Asymmetric Quantum Codes

The development of quantum error-correcting codes, although only ten years ofhistory, but has become a frontier of computer science、communications、physicsand mathematics, become a fast growing and challenging direction.This paper studies the classical error-correcting codes to construct quantumMDS codes and asymmetric quantum error-correcting codes. At last, three con-struction methods of quantum error-correcting codes were presented.This part of construct the quantum MDS codes, the main considerations onthe Hermitian space, by a fnite feld Fq2constructed Hermitian self-orthogonalcodes with the dual distance of4, where q is the odd prime power. Its main ideais that through the discussion of n in diferent ranges, respectively, to constructHermitian self-orthogonal codes generator matrix, to obtain the quantum greatdistance separable codes (MDS codes)with the minimum distance of4.This part of construct the asymmetric quantum error-correcting codes, In thispaper, frst, used the classical of Reed-Solomon codes to constructed a number ofasymmetric quantum error-correcting codes; second, to promote the concept of q-ary Reed-Muller codes based on the concept of binary Reed-Muller codes, and usedthe q-ary Reed-Muller codes constructed number of asymmetric quantum error-correcting codes; fnally, used the trace map of fnite feld extension feld to its ownsubdomain, the more asymmetric quantum error-correcting codes are obtained.