A new code concept is used for the L1 civil（L1C） signal of the global positioning system（GPS）.The generation of L1C codes is quite different from the generation of traditional ranging codes.Thus,it is necessary to...A new code concept is used for the L1 civil（L1C） signal of the global positioning system（GPS）.The generation of L1C codes is quite different from the generation of traditional ranging codes.Thus,it is necessary to find a method for the correct generation to pave the way for future research.L1C codes are based on only one Legendre sequence which consists of Legendre symbols.To calculate these Legendre symbols,the Euler criterion is always used to evaluate quadratic residues.However,due to the great length of L1C codes,this procedure causes overflow problems.Therefore,the quadratic reciprocity law,some related theorems and properties are introduced to solve the problems.Moreover,if the quadratic reciprocity law,some related theorems and properties are used to calculate different Legendre symbols,the combination modes may vary,which causes a complex generation process.The proposed generation method deals with this complex generation process effectively.In addition,through simulations,it is found that the autocorrelation features of obtained Legendre sequences and L1C codes are in accordance with theoretical results,which proves the correctness of the proposed method.展开更多
Let F_qbe afinite field with q=pmelements,where pis an odd prime and mis apositive integer.Here,let D_0={(x_1,x_2)∈F_q^2\{(0,0)}:Tr(x_1^(pk1+1)+x_2^(pk2+1))=c},where c∈F_q,Tr is the trace function fromFF_qtoFpand m/...Let F_qbe afinite field with q=pmelements,where pis an odd prime and mis apositive integer.Here,let D_0={(x_1,x_2)∈F_q^2\{(0,0)}:Tr(x_1^(pk1+1)+x_2^(pk2+1))=c},where c∈F_q,Tr is the trace function fromFF_qtoFpand m/(m,k_1)is odd,m/(m,k_2)is even.Define ap-ary linear code C_D =c(a_1,a_2):(a_1,a_2)∈F_q^2},where c(a_1,a_2)=(Tr(a_1x_1+a_2x_2))_((x1,x2)∈D).At most three-weight distributions of two classes of linear codes are settled.展开更多
The complexity of decoding the standard Reed-Solomon code is a well-known open problem in coding theory.The main problem is to compute the error distance of a received word.Using the Weil bound for character sum estim...The complexity of decoding the standard Reed-Solomon code is a well-known open problem in coding theory.The main problem is to compute the error distance of a received word.Using the Weil bound for character sum estimate,Li and Wan showed that the error distance can be determined when the degree of the received word as a polynomial is small.In the first part,the result of Li and Wan is improved.On the other hand,one of the important parameters of an error-correcting code is the dimension.In most cases,one can only get bounds for the dimension.In the second part,a formula for the dimension of the generalized trace Reed-Solomon codes in some cases is obtained.展开更多
基金supported by the National High Technology Research and Development Program of China(863Program)(2011AA110101)the National Basic Research Program of China(973Program)(2009CB724002)
文摘A new code concept is used for the L1 civil（L1C） signal of the global positioning system（GPS）.The generation of L1C codes is quite different from the generation of traditional ranging codes.Thus,it is necessary to find a method for the correct generation to pave the way for future research.L1C codes are based on only one Legendre sequence which consists of Legendre symbols.To calculate these Legendre symbols,the Euler criterion is always used to evaluate quadratic residues.However,due to the great length of L1C codes,this procedure causes overflow problems.Therefore,the quadratic reciprocity law,some related theorems and properties are introduced to solve the problems.Moreover,if the quadratic reciprocity law,some related theorems and properties are used to calculate different Legendre symbols,the combination modes may vary,which causes a complex generation process.The proposed generation method deals with this complex generation process effectively.In addition,through simulations,it is found that the autocorrelation features of obtained Legendre sequences and L1C codes are in accordance with theoretical results,which proves the correctness of the proposed method.
文摘Let F_qbe afinite field with q=pmelements,where pis an odd prime and mis apositive integer.Here,let D_0={(x_1,x_2)∈F_q^2\{(0,0)}:Tr(x_1^(pk1+1)+x_2^(pk2+1))=c},where c∈F_q,Tr is the trace function fromFF_qtoFpand m/(m,k_1)is odd,m/(m,k_2)is even.Define ap-ary linear code C_D =c(a_1,a_2):(a_1,a_2)∈F_q^2},where c(a_1,a_2)=(Tr(a_1x_1+a_2x_2))_((x1,x2)∈D).At most three-weight distributions of two classes of linear codes are settled.
基金Supported by the National Natural Science Foundation of China(61370089,61572168)the Anhui Provincial Natural Science Foundation(158085MA13)+1 种基金the Key Program of Excellent Youth Foundation of Higher Education in Anhui Province(2013SQRL071ZD)the National Natural Science Foundation for Young Scholars of Anhui Province(1508085QA04)
基金Project supported by the National Natural Science Foundation of China (No.10990011)the Doctoral Program Foundation of Ministry of Education of China (No.20095134120001)the Sichuan Province Foundation of China (No. 09ZA087)
文摘The complexity of decoding the standard Reed-Solomon code is a well-known open problem in coding theory.The main problem is to compute the error distance of a received word.Using the Weil bound for character sum estimate,Li and Wan showed that the error distance can be determined when the degree of the received word as a polynomial is small.In the first part,the result of Li and Wan is improved.On the other hand,one of the important parameters of an error-correcting code is the dimension.In most cases,one can only get bounds for the dimension.In the second part,a formula for the dimension of the generalized trace Reed-Solomon codes in some cases is obtained.
