The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacycli...The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.展开更多
In this paper,it has proposed a realtime implementation of low-density paritycheck(LDPC)decoder with less complexity used for satellite communication on FPGA platform.By adopting a(2048.4096)irregular quasi-cyclic(QC)...In this paper,it has proposed a realtime implementation of low-density paritycheck(LDPC)decoder with less complexity used for satellite communication on FPGA platform.By adopting a(2048.4096)irregular quasi-cyclic(QC)LDPC code,the proposed partly parallel decoding structure balances the complexity between the check node unit(CNU)and the variable node unit(VNU)based on min-sum(MS)algorithm,thereby achieving less Slice resources and superior clock performance.Moreover,as a lookup table(LUT)is utilized in this paper to search the node message stored in timeshare memory unit,it is simple to reuse and save large amount of storage resources.The implementation results on Xilinx FPGA chip illustrate that,compared with conventional structure,the proposed scheme can achieve at last 28.6%and 8%cost reduction in RAM and Slice respectively.The clock frequency is also increased to 280 MHz without decoding performance deterioration and convergence speed reduction.展开更多
In this paper, cyclic codes over the ring R = F ＋ uF ＋ vF ＋ uvF are discussed where the ring R is not a finite chain ring. By studying the polynomial ring Kn = （F ＋ uF ＋ vF ＋ uvF ）[x] / （x^n -1）and using the...In this paper, cyclic codes over the ring R = F ＋ uF ＋ vF ＋ uvF are discussed where the ring R is not a finite chain ring. By studying the polynomial ring Kn = （F ＋ uF ＋ vF ＋ uvF ）[x] / （x^n -1）and using the corresponding relationship between the cyclic codes in R and the ideals in Kn , cyclic codes over the ring R are characterized. Finally, a Gray-map is obtained and the image of cyclic codes in R is characterized.展开更多
In this article, we focus on cyclic and negacyclic codes of length 2p^s over the ring R = Fp^m ＋ uFp^m, where p is an odd prime. On the basis of the works of Dinh （in J.Algebra 324,940-950,2010）, we use the Chinese...In this article, we focus on cyclic and negacyclic codes of length 2p^s over the ring R = Fp^m ＋ uFp^m, where p is an odd prime. On the basis of the works of Dinh （in J.Algebra 324,940-950,2010）, we use the Chinese Remainder Theorem to establish the algebraic structure of cyclic and negacyclic codes of length 2p^s over the ring Fp^m ＋ uFp^m in terms of polynomial generators. Furthermore, we obtain the number of codewords in each of those cyclic and negacyclic codes.展开更多
The study of cyclic codes over rings has generated a lot of public interest.In this paper,we study cyclic codes and their dual codes over the ring Z P2 of length pe,and find a set of generators for these codes.The ran...The study of cyclic codes over rings has generated a lot of public interest.In this paper,we study cyclic codes and their dual codes over the ring Z P2 of length pe,and find a set of generators for these codes.The ranks and minimal generator sets of these codes are studied as well,which play an important role in decoding and determining the distance distribution of codes.展开更多
In this paper, we study skew cyclic codes over the ring Fp +vFp,where p is a odd prime and v 2=1. We give the generators of skew cyclic codes, with the consideration of the dual of skew cyclic codes.
In this paper, the period distribution of cyclic codes overR = F_q + uF_q +···+u^(m-1)F_q is studied, where um= 0 and q is a prime power. A necessary and sufficient condition for the existence of period...In this paper, the period distribution of cyclic codes overR = F_q + uF_q +···+u^(m-1)F_q is studied, where um= 0 and q is a prime power. A necessary and sufficient condition for the existence of period of cyclic codes over R is given. The period distributions of cyclic codes over R and their dual codes are determined by employing generator polynomial. The counting formulas of the period distributions of cyclic codes over R and their dual codes are obtained.展开更多
In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes...In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.展开更多
The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the co...The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors of n=p−12 length cyclic codes(C)over quaternion integers of Quaternion Mannheim(QM)weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengths n=p−12 and 2n−1=p−2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n=ϕ(p)=p−1 and length 2n−1=2ϕ(p)−1=2p−3(where p is prime integer andϕis Euler phi function)over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.展开更多
Let p =ef +1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e =3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, severa...Let p =ef +1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e =3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, several cyclotomic sequences of order e =3, 4, 6 are employed to construct a number of classes of cyclic codes over GF(q) with prime length. Under certain conditions, the linear complexity and reciprocal minimal polynomials of cyclotomic sequences are calculated, and the lower bounds on the minimum distances of these cyclic codes are obtained.展开更多
Let R be a finite chain ring with maximal ideal （7） and residue field F,and letγ be of nilpotency index t. To every code C of length n over R, a tower of codes C = （C ： γ0） C_ （C： 7） C ... C_ （C： γ2） C_ ...Let R be a finite chain ring with maximal ideal （7） and residue field F,and letγ be of nilpotency index t. To every code C of length n over R, a tower of codes C = （C ： γ0） C_ （C： 7） C ... C_ （C： γ2） C_ .-. C_ （C：γ^t-1） can be associated with C, where for any r C R, （C ： r） = {e C Rn I re E C}. Using generator elements of the projection of such a tower of codes to the residue field F, we characterize cyclic codes over R. This characterization turns the condition for codes over R to be cyclic into one for codes over the residue field F. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.展开更多
Let C be a free cyclic code over Zp^a and dim pC = k. In the paper, we prove that if the k characteristic generators of C are p-linearly independent then the corresponding nα- k characteristic generators of C^⊥ are ...Let C be a free cyclic code over Zp^a and dim pC = k. In the paper, we prove that if the k characteristic generators of C are p-linearly independent then the corresponding nα- k characteristic generators of C^⊥ are p-linearly independent. We then show that to any trellis that can be constructed from k p-linearly independent characteristic generators of C, there exists a trellis for C^⊥ with the same state-complexity profile, which generalizes the conjecture of Koetter and Vardy to a free cyclic code over Zpo.展开更多
We present the construction of quantum error-locating(QEL) codes based on classical error-locating(EL)codes. Similar to classical EL codes, QEL codes lie midway between quantum error-correcting codes and quantum error...We present the construction of quantum error-locating(QEL) codes based on classical error-locating(EL)codes. Similar to classical EL codes, QEL codes lie midway between quantum error-correcting codes and quantum errordetecting codes. Then QEL codes can locate qubit errors within one sub-block of the received qubit symbols but do not need to determine the exact locations of the erroneous qubits. We show that, an e-error-locating code derived from an arbitrary binary cyclic code with generator polynomial g(x), can lead to a QEL code with e error-locating abilities, only if g(x) does not contain the(1 + x)-factor.展开更多
In this paper,we investigate some classes of skew polycyclic codes and polycyclic codes over R=Z_(4)[u]/.We first obtain the generator polynomials of all(1,2u)-polycyclic codes over R.Then,by defining some Gray maps,w...In this paper,we investigate some classes of skew polycyclic codes and polycyclic codes over R=Z_(4)[u]/.We first obtain the generator polynomials of all(1,2u)-polycyclic codes over R.Then,by defining some Gray maps,we show that the images of(skew)(1,2u)-polycyclic codes over R are cyclic or quasi-cyclic with index 2 over Z_(4).Finally,an example of some(1,2u)-polycyclic codes over R is given to exhibit the main results of the paper.展开更多
In this paper, the cyclic code of the classic circuit is transformed and transplanted; then, the quantum encoding scheme based on cyclic code and quantum error-correction circuit is constructed. The proposed circuit c...In this paper, the cyclic code of the classic circuit is transformed and transplanted; then, the quantum encoding scheme based on cyclic code and quantum error-correction circuit is constructed. The proposed circuit can correct one-bit error, and the use of redundant bits to encode more than one-bit quantum information breaks the previous limitations of many bits encoding a quantum bit. Compared with the existing coding circuits （Shor code, Steane code and five stable subcode）, it shows obvious superiority in the quantum coding efficiency and transmission efficiency.展开更多
Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal...Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.展开更多
As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper,...As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.展开更多
A novel fast despreading scheme for M-ary Multi-Carrier Code-Division Multiple Access (MC-CDMA) system is proposed based on cyclic spreading codes and pre-equalizer. In the transmitter, the M spreading codes of each u...A novel fast despreading scheme for M-ary Multi-Carrier Code-Division Multiple Access (MC-CDMA) system is proposed based on cyclic spreading codes and pre-equalizer. In the transmitter, the M spreading codes of each user are generated by circularly shifting the prototype spreading code. A feedback pre-equalizer is employed to process the M-ary MC- CDMA signal before transmitted. The received signal is multiplied by the Inverse Discrete Fourier Transform (IDFT) result of the mirror image code of the prototype spreading code, and then demodulated by Orthogonal Frequency-Division Multiplexing (OFDM) demodulator. Compared with the conventional M-ary MC-CDMA receiver, the proposed scheme increases bandwidth efficiency, meanwhile, it achieves M-ary despread spectrum and multi-carrier demodulation, which reduces computation complexity remarkably.展开更多
For any odd prime p, a classification of all cyclic and negacyclic codes of length 8ps over Fpm are obtained, which establishes the algebra structures in term of specified polynomial generators of such codes. Among ot...For any odd prime p, a classification of all cyclic and negacyclic codes of length 8ps over Fpm are obtained, which establishes the algebra structures in term of specified polynomial generators of such codes. Among other results, all self-dual negacyclic codes of length 8ps are obtained, and the structures of α-constacyclic and β-constacyclic codes of length 8ps over Fpm are established.展开更多
In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe ...In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe Chapote, these maximal codes are constructed over finite fields, and these codes are used for coding and decoding.展开更多
基金Partly supported by the National Natural Science Foundations of China (No.60673074)key project of Ministry of Education Science and Technology’s Research (107065).
文摘The ranks of cyclic and negacyclic codes over the finite chain ring R as well as their minimal generating sets are defined, and then the expression forms we presented by studying the structures of cyclic and negacyclic codes over the finite chain ring R. Through the paper, it is assumed that the length of codes n can not be divided by the characteristic of R.
文摘In this paper,it has proposed a realtime implementation of low-density paritycheck(LDPC)decoder with less complexity used for satellite communication on FPGA platform.By adopting a(2048.4096)irregular quasi-cyclic(QC)LDPC code,the proposed partly parallel decoding structure balances the complexity between the check node unit(CNU)and the variable node unit(VNU)based on min-sum(MS)algorithm,thereby achieving less Slice resources and superior clock performance.Moreover,as a lookup table(LUT)is utilized in this paper to search the node message stored in timeshare memory unit,it is simple to reuse and save large amount of storage resources.The implementation results on Xilinx FPGA chip illustrate that,compared with conventional structure,the proposed scheme can achieve at last 28.6%and 8%cost reduction in RAM and Slice respectively.The clock frequency is also increased to 280 MHz without decoding performance deterioration and convergence speed reduction.
基金Supported by the Natural Science Foundation of Hubei Province (B20114410)
文摘In this paper, cyclic codes over the ring R = F ＋ uF ＋ vF ＋ uvF are discussed where the ring R is not a finite chain ring. By studying the polynomial ring Kn = （F ＋ uF ＋ vF ＋ uvF ）[x] / （x^n -1）and using the corresponding relationship between the cyclic codes in R and the ideals in Kn , cyclic codes over the ring R are characterized. Finally, a Gray-map is obtained and the image of cyclic codes in R is characterized.
基金supported by the Natural ScienceFoundation of Hubei Province(D2014401)the Natural Science Foundation of Hubei Polytechnic University(12xjz14A)
文摘In this article, we focus on cyclic and negacyclic codes of length 2p^s over the ring R = Fp^m ＋ uFp^m, where p is an odd prime. On the basis of the works of Dinh （in J.Algebra 324,940-950,2010）, we use the Chinese Remainder Theorem to establish the algebraic structure of cyclic and negacyclic codes of length 2p^s over the ring Fp^m ＋ uFp^m in terms of polynomial generators. Furthermore, we obtain the number of codewords in each of those cyclic and negacyclic codes.
基金the National Natural Science Foundation of China(No.60673074)the Key Project of Ministry of Education Science and Technology’s Research(107065)
文摘The study of cyclic codes over rings has generated a lot of public interest.In this paper,we study cyclic codes and their dual codes over the ring Z P2 of length pe,and find a set of generators for these codes.The ranks and minimal generator sets of these codes are studied as well,which play an important role in decoding and determining the distance distribution of codes.
基金Supported by the National Natural Science Foundation of China(No.61370089)
文摘In this paper, we study skew cyclic codes over the ring Fp +vFp,where p is a odd prime and v 2=1. We give the generators of skew cyclic codes, with the consideration of the dual of skew cyclic codes.
基金Supported by the National Natural Science Foundation of China(No.61370089)Fundamental Research Funds for the Central Universities(Nos.2013HGCH0024,J2014HGXJ0073)Specialized Research Fund for the Doctoral Program of Hefei University of Technology(No.JZ2014HGBZ0029)
文摘In this paper, the period distribution of cyclic codes overR = F_q + uF_q +···+u^(m-1)F_q is studied, where um= 0 and q is a prime power. A necessary and sufficient condition for the existence of period of cyclic codes over R is given. The period distributions of cyclic codes over R and their dual codes are determined by employing generator polynomial. The counting formulas of the period distributions of cyclic codes over R and their dual codes are obtained.
基金supported by SRF for ROCS,SEM,the Key Project of Chinese Ministry of Education (108099)CCNU Project (CCNU09Y01003)
文摘In this article, cyclic codes and negacyclic codes over formal power series rings are studied. The structure of cyclic codes over this class of rings is given, and the relationship between these codes and cyclic codes over finite chain rings is obtained. Using an isomorphism between cyclic and negacyclic codes over formal power series rings, the structure of negacyclic codes over the formal power series rings is obtained.
基金The authors extend their gratitude to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number R.G.P.1/85/42.
文摘The decoding algorithm for the correction of errors of arbitrary Mannheim weight has discussed for Lattice constellations and codes from quadratic number fields.Following these lines,the decoding algorithms for the correction of errors of n=p−12 length cyclic codes(C)over quaternion integers of Quaternion Mannheim(QM)weight one up to two coordinates have considered.In continuation,the case of cyclic codes of lengths n=p−12 and 2n−1=p−2 has studied to improve the error correction efficiency.In this study,we present the decoding of cyclic codes of length n=ϕ(p)=p−1 and length 2n−1=2ϕ(p)−1=2p−3(where p is prime integer andϕis Euler phi function)over Hamilton Quaternion integers of Quaternion Mannheim weight for the correction of errors.Furthermore,the error correction capability and code rate tradeoff of these codes are also discussed.Thus,an increase in the length of the cyclic code is achieved along with its better code rate and an adequate error correction capability.
基金Supported by the National Natural Science Foundation(NNSF)of China(No.11171150)Foundation of Science and Technology on Information Assurance Laboratory(No.KJ-13-001)+1 种基金Funding of Jiangsu Innovation Program for Graduate Education(CXLX13-127,Fundamental Research Funds for the Central Universities)Funding for Outstanding Doctoral Dissertation in NUAA(BCXJ-13-17)
文摘Let p =ef +1 be an odd prime with positive integers e and f. In this paper, we calculate the values of Gauss periods of order e =3, 4, 6 over a finite field GF(q), where q is a prime with q≠p. As applications, several cyclotomic sequences of order e =3, 4, 6 are employed to construct a number of classes of cyclic codes over GF(q) with prime length. Under certain conditions, the linear complexity and reciprocal minimal polynomials of cyclotomic sequences are calculated, and the lower bounds on the minimum distances of these cyclic codes are obtained.
基金supported by the Natural Science Foundation of Hubei Province (B20114410)the Natural Science Foundation of Hubei Polytechnic University (12xjz14A)
文摘Let R be a finite chain ring with maximal ideal （7） and residue field F,and letγ be of nilpotency index t. To every code C of length n over R, a tower of codes C = （C ： γ0） C_ （C： 7） C ... C_ （C： γ2） C_ .-. C_ （C：γ^t-1） can be associated with C, where for any r C R, （C ： r） = {e C Rn I re E C}. Using generator elements of the projection of such a tower of codes to the residue field F, we characterize cyclic codes over R. This characterization turns the condition for codes over R to be cyclic into one for codes over the residue field F. Furthermore, we obtain a characterization of cyclic codes over the formal power series ring of a finite chain ring.
基金Supported by the National Natural Science Foundation of China(60673071)
文摘Let C be a free cyclic code over Zp^a and dim pC = k. In the paper, we prove that if the k characteristic generators of C are p-linearly independent then the corresponding nα- k characteristic generators of C^⊥ are p-linearly independent. We then show that to any trellis that can be constructed from k p-linearly independent characteristic generators of C, there exists a trellis for C^⊥ with the same state-complexity profile, which generalizes the conjecture of Koetter and Vardy to a free cyclic code over Zpo.
基金Supported by the National Natural Science Foundation of China under Grant No.61170321the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20110092110024the Scientific Research Innovation Plan for College Graduates of Jiangsu Province under Grant No.CXZZ13 0105
文摘We present the construction of quantum error-locating(QEL) codes based on classical error-locating(EL)codes. Similar to classical EL codes, QEL codes lie midway between quantum error-correcting codes and quantum errordetecting codes. Then QEL codes can locate qubit errors within one sub-block of the received qubit symbols but do not need to determine the exact locations of the erroneous qubits. We show that, an e-error-locating code derived from an arbitrary binary cyclic code with generator polynomial g(x), can lead to a QEL code with e error-locating abilities, only if g(x) does not contain the(1 + x)-factor.
基金Supported by the National Natural Science Foundation of China(Grant No.12201361)。
文摘In this paper,we investigate some classes of skew polycyclic codes and polycyclic codes over R=Z_(4)[u]/.We first obtain the generator polynomials of all(1,2u)-polycyclic codes over R.Then,by defining some Gray maps,we show that the images of(skew)(1,2u)-polycyclic codes over R are cyclic or quasi-cyclic with index 2 over Z_(4).Finally,an example of some(1,2u)-polycyclic codes over R is given to exhibit the main results of the paper.
基金Supported by the National Natural Science Foundation of China (61271122)the Natural Science Foundation of Anhui Province(1208085MF102)
文摘In this paper, the cyclic code of the classic circuit is transformed and transplanted; then, the quantum encoding scheme based on cyclic code and quantum error-correction circuit is constructed. The proposed circuit can correct one-bit error, and the use of redundant bits to encode more than one-bit quantum information breaks the previous limitations of many bits encoding a quantum bit. Compared with the existing coding circuits （Shor code, Steane code and five stable subcode）, it shows obvious superiority in the quantum coding efficiency and transmission efficiency.
文摘Cyclic codes form an important class of codes. They have very interesting algebraic structure. Furthermore, they are equivalent to many important codes, such as binary Hamming codes, Golay codes and BCH codes. Minimal codewords in linear codes are widely used in constructing decoding algorithms and studying linear secret sharing scheme. In this paper, we show that in the binary cyclic code all of the codewords are minimal, except 0 and 1. Then, we obtain a result about the number of minimal codewords in the binary cyclic codes.
文摘As a branch of applied mathematics, coding theory plays an important role. Among them, cyclic codes have attracted much attention because of their good algebraic structure and easy analysis performance. In this paper, we will study one class of cyclic codes over F<sub>3</sub>. Given the length and dimension, we show that it is optimal by proving its minimum distance is equal to 4, according to the Sphere Packing bound.
基金Supported by the National Natural Science Foundation of China (No.60172029).
文摘A novel fast despreading scheme for M-ary Multi-Carrier Code-Division Multiple Access (MC-CDMA) system is proposed based on cyclic spreading codes and pre-equalizer. In the transmitter, the M spreading codes of each user are generated by circularly shifting the prototype spreading code. A feedback pre-equalizer is employed to process the M-ary MC- CDMA signal before transmitted. The received signal is multiplied by the Inverse Discrete Fourier Transform (IDFT) result of the mirror image code of the prototype spreading code, and then demodulated by Orthogonal Frequency-Division Multiplexing (OFDM) demodulator. Compared with the conventional M-ary MC-CDMA receiver, the proposed scheme increases bandwidth efficiency, meanwhile, it achieves M-ary despread spectrum and multi-carrier demodulation, which reduces computation complexity remarkably.
基金Supported by the Natural Science Foundation of Hubei Province(D20144401)the Natural Science Foundation of Hubei Polytechnic University(12xjz14A)
文摘For any odd prime p, a classification of all cyclic and negacyclic codes of length 8ps over Fpm are obtained, which establishes the algebra structures in term of specified polynomial generators of such codes. Among other results, all self-dual negacyclic codes of length 8ps are obtained, and the structures of α-constacyclic and β-constacyclic codes of length 8ps over Fpm are established.
文摘In this work we try to introduce the concept of Maximal codes that are built over rings, more precisely we will give Maximal codes for special rings, Namely that the notion of maximal codes has been used by Chritophe Chapote, these maximal codes are constructed over finite fields, and these codes are used for coding and decoding.
