Research On The Related Problems And Application Of BCH Codes

With the rapid development of quantum information technology,it has brought new opportunities and challenges to information science researches.The reliability,security and integrity of the information transmission and storage have raised the status of the information security research.As an important part of the information security theory,the theory of error correcting code plays a decisive role in ensuring the authenticity and reliability of communication.As a classic code with ingenious circulatory structure,BCH code has been developed rapidly.To construct the BCH codes with designed distances as large as possible and to characterize its dual containing conditions are the basis of quantum error correcting codes.Meanwhile,because of its strict algebraic structure,BCH code has been expanded its applications in image processing and compressed sensing.On the one hand,based on the dual containing condition and defining set of BCH code,this thesis focuses on the construction of stabilizer quantum error-correcting codes and asymmetric quantum error-correcting codes.On the other hand,this thesis mainly solves the practical application of generator matrix and parity check matrix of BCH codes in Locality-Sensitive Hashing algorithm and compressed sensing.The major contributions and innovations of this thesis are as following:1.Constructions of stabilizer quantum error-correcting codes with Hermite dual containing BCH codesFirstly,by a careful analysis on properties of cyclotomic cosets in the defining sets of imprimitive narrow-sense BCH codes with six families of code lengths over finite field(? 3 is prime power)respectively,we determine their new maximal designed distance for Hermite dual containing condition and prove that the new maximal designed-distance can be much larger than the result obtained by the method given in previous literature for each different code length correspondingly.Secondly,we solve the dimension calculation problem of these narrow-sense imprimitive BCH codes,thus six families of stabilizer quantum error-correcting codes with good parameters are constructed via CSS construction,the newly obtained quantum BCH codes have larger distance when compared with the known results in previous literatures.2.Constructions of asymmetric quantum error-correcting codes with pairs of nested BCH codesFirstly,from the perspective of properties of cyclotomic cosets in the defining sets of pairs of nested BCH codes,we take a more precise exploration on the dual containing condition of two nested BCH codes,and construct three familes of primitive narrow-sense BCH codes over finite field(? 3 is an arbitrary prime power).Secondly,under the Hermite inner product operation,the sufficient and necessary conditions for dual containing pairs of nested BCH codes are analyzed,and the parameters of pairs of nested BCH code over are determined,respectively.Finally,we construct new asymmetric quantum error-correcting codes with good parameters from these familes of pairs of nested BCH codes.Especially,real-distance of our asymmetric quantum codes,which are much larger than the result obtained in previous literature,and these conclusions improve the error-correction ability of quantum codes.3.Constructions of hash function with generator matrix of BCH codesFirstly,based on-ary BCH codes,the relationship between the parameters of BCH codes and collision probability is established.Secondly,according to the characteristics of the designed distance limited for-ary BCH code,BCHp-LSH algorithm uses the generator matrix of BCH codes as a transform basis of the hash function to mapping the source data into the hash space.Finally,to illustrate the advantages of our algorithm,an experimental system of BCHp-LSH is developed.Then,by Patches dataset,Simulation dataset and Surveillance dataset,we conduct multiple comparison experiments between E2 LSH and our BCHp-LSH.The experiments results show that the BCHp-LSH algorithm is superior to E2 LSH algorithm in average accuracy,average recall ratio and running speed.Therefore,our BCHp-LSH algorithm achieves both feasible and practical value.4.Constructions of measurement matrix with parity check matrix of BCH codesFirstly,the structure characteristics of the parity check matrix of dual containing BCH codes are analyzed,and the conditions for constructional detail of the measurement matrix are discussed.Secondly,the relationship between the parameters of the BCH code and the correlation coefficient of the measurement matrix is established.Moreover,under the condition of constant compression ratio,the purpose for reducing the value of correlation coefficient between the measurement matrix and the sparse orthogonal matrix is realized.All in all,the study provides support for high probability reconstruction of signals,and it provides a good idea for the construction of measurement matrix for compressed sensing.