In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the s...In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the Mac Williams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.展开更多
Abraham Lempel et al made a connection between linear codes and systems of bilinear forms over finite fields. In this correspondence, a new simple proof of a theorem in [1] is presented; in addition, the encoding proc...Abraham Lempel et al made a connection between linear codes and systems of bilinear forms over finite fields. In this correspondence, a new simple proof of a theorem in [1] is presented; in addition, the encoding process and the decoding procedure of RS codes are simplified via circulant matrices. Finally, the results show that the correspondence between bilinear forms and linear codes is not unique.展开更多
The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved i...The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with heuristic techniques such as genetic algorithms and local search algorithms. In this paper we propose two approaches to attack the hardness of this problem. The first approach is based on genetic algorithms and it yield to good results comparing to another work based also on genetic algorithms. The second approach is based on a new randomized algorithm which we call 'Multiple Impulse Method (MIM)', where the principle is to search codewords locally around the all-zero codeword perturbed by a minimum level of noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum Hamming-weight codeword whose Hamming weight is equal to the minimum distance of the linear code.展开更多
In this paper,generalized sparse(GS)codes are proposed to support reliable and efficient transmission over non-Gaussian channels.Specifically,by expanding the single-parity check(SPC)code constraints with powerful alg...In this paper,generalized sparse(GS)codes are proposed to support reliable and efficient transmission over non-Gaussian channels.Specifically,by expanding the single-parity check(SPC)code constraints with powerful algebraic codes,GS codes generalize conventional sparse codes with enhanced error-correcting capability,as well as better code design flexibility by covering a wide range of block-lengths and coding rates with reduced encoding/decoding complexity.Moreover,by introducing a universal communication channel model,a general framework for performance analysis and code design of GS codes is formulated,by which the coding parameters can be optimized for different target channel conditions.Finally,example codes are constructed for several critical application scenarios with non-Gaussian channels.Numerical simulations are performed to demonstrate the superiority of the proposed GS coding scheme to traditional channel coding schemes.展开更多
Given the potential for bit flipping of data on a memory medium,a high-speed parallel Bose-Chaudhuri-Hocquenghem(BCH)error correction scheme with modular characteristics,combining logic implementation and a look-up ta...Given the potential for bit flipping of data on a memory medium,a high-speed parallel Bose-Chaudhuri-Hocquenghem(BCH)error correction scheme with modular characteristics,combining logic implementation and a look-up table,is proposed.It is suitable for data error correction on a modern field programmable gate array full with on-chip embedded memories.We elaborate on the optimization method for each part of the system and analyze the realization process of this scheme in the case of the BCH code with an information bit length of 1024 bits and a code length of 1068 bits that corrects the 4-bit error.展开更多
In this study, a class of Generalized Low-Density Parity-Check (GLDPC) codes is designed for data transmission over a Partial-Band Jamming (PBJ) environment. The GLDPC codes are constructed by replacing parity-che...In this study, a class of Generalized Low-Density Parity-Check (GLDPC) codes is designed for data transmission over a Partial-Band Jamming (PBJ) environment. The GLDPC codes are constructed by replacing parity-check code constraints with those of nonsystematic Bose-Chaudhuri-Hocquenghem (BCH), referred to as Low-Density Parity-Check (LDPC)-BCH codes. The rate of an LDPC-BCH code is adjusted by selecting the transmission length of the nonsystematic BCH code, and a low-complexity decoding algorithm based on message- passing is presented that employs A Posteriori Probability (APP) fast BCH transform for decoding the BCH check nodes at each decoding iteration. Simulation results show that the LDPC-BCH codes with a code rate of 1/8.5 have a bit error rate performance of 1 x10-8 at signal-noise-ratios of -6.97 dB, -4.63 dB, and 2.48 dB when the fractions of the band jammed are 30%, 50%, and 70%, respectively.展开更多
基金the National Natural Science Foundation of China (Grant Nos. 61972413, 61901525, and 62002385)the National Key R&D Program of China (Grant No. 2021YFB3100100)RGC under Grant No. N HKUST619/17 from Hong Kong, China。
文摘In a recent paper, Hu et al. defined the complete weight distributions of quantum codes and proved the Mac Williams identities, and as applications they showed how such weight distributions may be used to obtain the singleton-type and hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the Mac Williams identities of the complete weight distributions of quantum codes. We obtain new information about the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes. We get new identities involving the complete weight distributions of two different quantum codes. We estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.
基金She was with the Department of Mathematics in Wuhan University while writting this paper.
文摘Abraham Lempel et al made a connection between linear codes and systems of bilinear forms over finite fields. In this correspondence, a new simple proof of a theorem in [1] is presented; in addition, the encoding process and the decoding procedure of RS codes are simplified via circulant matrices. Finally, the results show that the correspondence between bilinear forms and linear codes is not unique.
文摘The evaluation of the minimum distance of linear block codes remains an open problem in coding theory, and it is not easy to determine its true value by classical methods, for this reason the problem has been solved in the literature with heuristic techniques such as genetic algorithms and local search algorithms. In this paper we propose two approaches to attack the hardness of this problem. The first approach is based on genetic algorithms and it yield to good results comparing to another work based also on genetic algorithms. The second approach is based on a new randomized algorithm which we call 'Multiple Impulse Method (MIM)', where the principle is to search codewords locally around the all-zero codeword perturbed by a minimum level of noise, anticipating that the resultant nearest nonzero codewords will most likely contain the minimum Hamming-weight codeword whose Hamming weight is equal to the minimum distance of the linear code.
基金This work was supported by the National Natural Science Foundation of China(Grants No.62025110 and 62101308).
文摘In this paper,generalized sparse(GS)codes are proposed to support reliable and efficient transmission over non-Gaussian channels.Specifically,by expanding the single-parity check(SPC)code constraints with powerful algebraic codes,GS codes generalize conventional sparse codes with enhanced error-correcting capability,as well as better code design flexibility by covering a wide range of block-lengths and coding rates with reduced encoding/decoding complexity.Moreover,by introducing a universal communication channel model,a general framework for performance analysis and code design of GS codes is formulated,by which the coding parameters can be optimized for different target channel conditions.Finally,example codes are constructed for several critical application scenarios with non-Gaussian channels.Numerical simulations are performed to demonstrate the superiority of the proposed GS coding scheme to traditional channel coding schemes.
基金Project supported by the National Natural Science Foundation of China(No.61973280)the China Postdoctoral Science Foundation(No.2019M661069)。
文摘Given the potential for bit flipping of data on a memory medium,a high-speed parallel Bose-Chaudhuri-Hocquenghem(BCH)error correction scheme with modular characteristics,combining logic implementation and a look-up table,is proposed.It is suitable for data error correction on a modern field programmable gate array full with on-chip embedded memories.We elaborate on the optimization method for each part of the system and analyze the realization process of this scheme in the case of the BCH code with an information bit length of 1024 bits and a code length of 1068 bits that corrects the 4-bit error.
基金supported by the National Natural Science Foundation of China (Nos. 61101072 and 61132002)
文摘In this study, a class of Generalized Low-Density Parity-Check (GLDPC) codes is designed for data transmission over a Partial-Band Jamming (PBJ) environment. The GLDPC codes are constructed by replacing parity-check code constraints with those of nonsystematic Bose-Chaudhuri-Hocquenghem (BCH), referred to as Low-Density Parity-Check (LDPC)-BCH codes. The rate of an LDPC-BCH code is adjusted by selecting the transmission length of the nonsystematic BCH code, and a low-complexity decoding algorithm based on message- passing is presented that employs A Posteriori Probability (APP) fast BCH transform for decoding the BCH check nodes at each decoding iteration. Simulation results show that the LDPC-BCH codes with a code rate of 1/8.5 have a bit error rate performance of 1 x10-8 at signal-noise-ratios of -6.97 dB, -4.63 dB, and 2.48 dB when the fractions of the band jammed are 30%, 50%, and 70%, respectively.