| Information has been all over every aspect of the modern society.Information can bring convenience to people.But inevitably there are some shortcomings during infor-mation acquisition,transmission,storage and other processes because of noise and other factors.How to improve the reliability and effectiveness of information transmission is always an important goal in communication work.Error-correction code technology is an important means to improve the reliability of information transmission.In the infor-mation age today,error-correction code has become a standard technology in the field of communication.However,in error correction code technology,how to improve the capability of error-detection and error-correction of codes is a very critical problem.Of the error-detection,error-correction is an important indicator of the Hamming weight.Though the research of Hamming weight is a.basic theory problem,it has important application value and practical value.Hamming is one of the founders of coding theory,he proposed the Hamming weight.In 1991,V.K.Wei proposed the concept of generalized Hamming weight,of linear codes as he studied the cryptographic properties of the Type II stealthy channel.When the linear code C with parameter[n,k;q]was used to the Type II stealthy channel,if the enemy wants to get r bits from s bytes,bytes proved s>dr,where dr is the minimum distance of the r dimension of the code C.Therefore the Hamming weight of code C Fully describes its characteristic.In 1996;Chen Wende and T.Kl(?)ve proposed the finite projective geometry method,and this method was first effectively used to determine the weight hierarchies of q-ary linear codes with dimension 4.And they proposed the concept of assignment function.For the three-dimensional and 4-dimensional linear codes,the almost all weight.hierarchies have been determined by the finite projective geometry.The finite projective geometry is effective to study the weight hierarchies for low dimension(the dimension k<4)and q=2,3.As the dimension increases,it is very difficult to determine the weight hierarchies of linear codes.How to continue to use this theory to determine the weight hierarchies of the high dimension is a worthwhile question.The technology of error-correction code is not,only important,for the classical code,but also plays an important role in quantum information theory.With the emergence of quantum computer and the development of quantum theory,quantum information is attracting more and more attention.All the advantages of quantum computers come from quantum coherence.However,in the real world,the quantum bits are not isolated,they interact with the external environment at all times,and the quantum coherence will inevitably occur.The quantum coherence of the quantum computer will be independent.of the quantum coherence.The exponential decay occurs over time,causing quantum decoherence,quantum decoherence to cause quantum errors,and the possible benefits of using quantum state encoding information.Therefore,how to reduce or avoid these quantum errors in quantum computers,and how to detect and correct quantum errors are very important.Quantum error-correction code theory has long been regarded as the main method of counteracting the coherence effect of quantum,which is the key technology to realize the actual quantum communication and quantum computation.Based on the above research background,we study the general linear codes with dimension 5,focuses on the class VI.And this thesis generalizes the study of classical error-correction codes to quantum error-correction codes.The preliminary research on quantum error correction code.is carried out and a quantum error-correction in GF(2)is determined.This paper studies the VI class of the Hamming weight hierarchies of q-ary linear codes with dimension 5.According to this type of existing necessary conditions,some new constraints can be found.And according to these new constraints and the necessary conditions of this class,the VI class can be re-classified.The class VI can be divided into six small sub-categories,and they are VI-1,VI-2,VI-3,VI-4,VI-5 and VI-6.The necessary condition of every sub-category is proposed.Therefore,we can research the six sub-classes.As long as almost all Hamming weight hierarchies of each subclass are determined,almost all Hamming weight hierarchies of class VI are determined.In summary,the main conclusions are as follows:(1)The VI class of the Hamming weight hierarchies of q-ary linear codes with dimension 5 is reviewed.According to this type of exist.ing necessary conditions,some new constraints can be found.And according t.o these new constraints and the necessary conditions of this class,the VI class can be re-classified.The class VI can be divided into six small sub-categories,and they are VI-1,VI-2,VI-3,VI-4,VI-5 and VI-6.The necessary condition of every sub-category is proposed.And we study the first sub-class of the six sub-classes.By using the finite projective geometry method,we determine nearly all the weight hierarchies of the VI-1 subclass of weight hierarchies of linear codes with dimension 5.(2)The second subclass of class VI is studied;and almost all of the Hamming weight hierarchies of VI-2 are determined by projection to finite projective space,and the necessary conditions for this class are proved to be almost.sufficient..(3)The fifth subclass of class VI is studied,and almost all Hamming weight hier-archies of VI-5 are determined by the met.hod of finite projective geometry.Combined with the necessary conditions of this subclass-and the requirements of class VI,we prove that the necessary conditions for this class are almost sufficient.(4)We generalizes the study of classical error-correction codes to quantum error-correction codes.The preliminary research on quantum error correction code is carried out and a quantum error-correction in GF(2)is determined... |
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