Construction Of M-MDS Codes Over Finite Fields

With the rapid development of communication technology and information industry,various communication modes are emerging and widely used,and the storage and exchange of digital information are also increasing day by day.In the process of digital signal transmission in the channel,channel noise may damage the transmitted data.So the error code often occurs at the receiving end.In order to enable the receiver to check and correct the errors and then restore the true information,error correction codes emerge as the times require At present,error correction coding technology has been widely used in modern cryptography,satellite communications,mobile communications,computer data storage and transmission,image recording,digital audio and broadcasting and so on.In the era of network information,the error correction coding will be applied much extensivelyThe MDS code refers to a code whose parameter[n,k,d]satisfies the equal conditions of Singleton boundary n-k-d + 1?0.When the parameters n and k are given,the maximal distance separable code with code length n and information bit k is the code with the strongest error correction ability.It has been praised as the most attractive error-correcting code by famous scholars F.J.Macwilliams and N.J.A.Sloane.Although MDS codes are considered to be ideal linear codes,long MDS codes are very few,which drives us look for new similar codes.Later,inspired by V.K.Wei's research on generalized Hamming weight,S.M.Dodunekov and I.N.Landgev defined and studied NMDS(1-MDS)codes by weakening the definition restriction of MDS codes so that the parameters satisfy S(C)=S(C?)= n-k-d + 1 = 1.Then Hongxi Tong further defined and studied NNMDS(2-MDS)codes by weakening the definition restriction of MDS codes so that the parameters satisfy S(C)=S(C?)= n-k-d + 1 = 2.Finally,Qunying Liao unified definition and study the extended form of MDS code m-MDS codeIn view of the fact that the data processed by the computer is in the form of 0-1,we focus on the concrete construction of m-MDS codes in the finite field F2.Starting from the theoretical sufficient and necessary conditions of m-MDS codes,we can get the range of parameter m ?5 by analyzing the relationship between the number and location of 0 and 1 in the check matrix of m-MDS codes.Then we construct a series of check matrices of m-MDS codes by using MATLAB language program.Thus we obtain a series of the encoding and decoding algorithms of m-MDS codes.