New Quantum MDS Codes From GRS Codes

It is an important task to construct the quantum codes with good parameters in the study of quantum error-correcting codes.The quantum codes have two important parameters,the length and the minimum distance of the quantum codes.Quantum MDS codes are a special class of codes satisfying the quantum Singleton bound.They have a strong error-correcting ability.In addition,they have good algebraic structure,convenient construction and can be also encoded and decoded relatively easily.In recent years,many scholars have constructed quantum MDS codes by means of known classical errorcorrecting codes,such as BCH codes,algebraic geometry codes and constacylic codes,etc.The classes of GRS codes(Generalized Reed-Solomon Codes)are also effective tools for constructing quantum MDS codes with good performance.Based on the GRS codes,two new classes of quantum MDS codes on a finite field Fq2 are constructed in this dissertation,whose minimum distances are greater than q/2+1 under certain conditions.At first,the parameter q is considered by divided into two cases:q=2mand q=pm(where p is a prime number greater than 2).These cases are discussed respectively and the new quantum MDS codes of length n are constructed.The quantum MDS codes whose minimum distances are greater than q/2+1 are found by giving the corresponding L1-forms and L2-forms.Secondly,a class of new quantum MDS codes of length 1+n are constructed by lengthening the vectors.At last,we find out the quantum MDS codes whose minimum distances are greater than q/2+1 by discussing further the corresponding equations.The results in this dissertation extend the previous research work and is helpful to study forward the quantum MDS codes in the later.