The Research On The Error-correcting Pair For Several Classes Of Linear Codes

In decoding theory,there are usually three principles for decoding lin-ear codes over finite fields,namely,unambiguous decoding,list decoding and maximum-likelihood decoding.In particular,the decoding method based on the error-correcting pair is an unambiguous decoding method for linear codes,which exists for many classical lin-ear codes such as generalized Reed-Solomon（in short,GRS）codes,cyclic codes,algebraic geometry codes,etc..For any linear codeover the finite field F,if an error-correcting pair ofcan be constructed,then a decoding algorithm foris obtained,therefore,the study of the existence and constructions of the error-correcting pair for linear codes is in-teresting.It is well-known that every linear codeis contained in an Maximal Distance Separable（in short,MDS）linear code with the same minimum distance as that ofover some finite field extensions,and in recent years,twisted generalized Reed Solomon（in short,TGRS）codes,have been attracted attentions due to their excellent algebraic proper-ties and application values,but a few results have been given for the decoding.On the other hand,one of the most important MDS codes is the GRS code.In this thesis,we define a class of linear codes–lengthened GRS codes which are not equivalent to GRS codes and have excellent algebraic properties:it is either MDS or Almost-MDS（in short,AMDS）linear codes.Furthermore,the direct sum of linear codes is one of the important method for constructing new codes,thus discussing the error-correcting pair of the direct sum code is also significant.In summary,we mainly give the following three parts of results.Part 1 We discuss the parameters of the error-correcting pair for MDS linear codes with even minimum distance,and obtain the characteristics of MDS linear codes with the error-correcting pair and an even minimum distance.Firstly,we show that for an MDS linear codewith an even minimum distance and an error-correcting pair（A,（?））,the parameters ofare three cases.Secondly,for one case,a necessary condition for thathas an error-correcting pair is given,namely,is a GRS code.Finally,for the other two cases,we give some counterexamples for thatis a non-GRS code.Part 2 We study the existence and constructions of the error-correcting pair for both TGRS codes and lengthened GRS codes.For TGRS codes,firstly,we show that several classes of TGRS codes are not equivalent to GRS codes.Secondly,a necessary and suffi-cient condition for a TGRS code with=2 to be MDS,AMDS or Almost-Almost-MDS（in short,AAMDS）linear codes are given,respectively.Finally,several sufficient condi-tions for a TGRS code with=1 or 2 and an error-correcting pair or not are given.For lengthened GRS codes,firstly,several sufficient conditions for that lengthened GRS codes are not equivalent to GRS codes are given.Secondly,a necessary and sufficient condition for a lengthened GRS code to be an MDS or AMDS linear code is given,respectively.Finally,several sufficient conditions for a lengthened GRS code with an error-correcting pair or not are given.Part 3 We study the existence and constructions of the error-correcting pair for the direct sum code of two linear codes with the error-correcting pair.Firstly,we give the possible values of the parameters for an error-correcting pair of an NMDS linear code.Secondly,for the direct sum codeof two linear codes with an error-correcting pair,several sufficient conditions forwith an error-correcting pair are given.Finally,for the direct sum codeof two MDS linear codes,two NMDS linear codes,or an MDS linear code and an NMDS linear code,several sufficient conditions forwith an error-correcting pair are given,respectively.