Construction and Weight Distributions of Binary Linear Codes Based on Deep Holes

Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.

Homomorphic Error-Control Codes for Linear Network Coding in Packet Networks

In this work, the homomorphism of the classic linear block code in linear network coding for the case of binary field and its extensions is studied. It is proved that the classic linear error-control block code is homomorphic network error-control code in network coding. That is, if the source packets at the source node for a linear network coding are precoded using a linear block code, then every packet flowing in the network regarding to the source satisfies the same constraints as the source. As a consequence, error detection and correction can be performed at every intermediate nodes of multicast flow, rather than only at the destination node in the conventional way, which can help to identify and correct errors timely at the error-corrupted link and save the cost of forwarding error-corrupted data to the destination node when the intermediate nodes are ignorant of the errors. In addition, three examples are demonstrated which show that homomorphic linear code can be combined with homomorphic signature, McEliece public-key cryptosystem and unequal error protection respectively and thus have a great potential of practical utility.

On Latency Reductions in Vehicle-to-Vehicle Networks by Random Linear Network Coding

In the Internet of vehicles(IoV),direct communication between vehicles,i.e.,vehicle-tovehicle(V2V)may have lower latency,compared to the schemes with help of Road Side Unit(RSU)or base station.In this paper,the scenario where the demands of a vehicle are satisfied by cooperative transmissions from those one in front is considered.Since the topology of the vehicle network is dynamic,random linear network coding is applied in such a multisource single-sink vehicle-to-vehicle network,where each vehicle is assumed to broadcast messages to others so that the intermediate vehicles between sources and sink can reduce the latency collaboratively.It is shown that the coding scheme can significantly reduce the time delay compared with the non-coding scheme even in the channels with high packet loss rate.In order to further optimize the coding scheme,one can increase the generation size,where the generation size means the number of raw data packets sent by the source node to the sink node in each round of communication.Under the premise of satisfying the coding validity,we can dynamically select the Galois field size according to the number of intermediate nodes.It is not surprised that the reduction in the Galois field size can further reduce the transmission latency.

New algorithm for variable-rate linear broadcast network coding

To adjust the variance of source rate in linear broadcast networks,global encoding kernels should have corresponding dimensions to instruct the decoding process.The algorithm of constructing such global encoding kernels is to adjust heterogeneous network to possible link failures.Linear algebra,graph theory and group theory are applied to construct one series of global encoding kernels which are applicable to all source rates.The effectiveness and existence of such global encoding kernels are proved.Based on information flow,the algorithm of construction is explicitly given within polynomial time(|E·|T|ω2max) and the memory complexity of algorithm is O(|E|).Both time and memory complexity of this algorithm proposed can be O(ωmax) less than those of algorithms in related works.

BILINEAR FORMS AND LINEAR CODES

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.

Linear Codes with Two Weights or Three Weights from Two Types of Quadratic forms

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.

Optimal and suboptimal structured algorithms of binary linear block codes

The optimal and suboptimal structured algorithms of linear block codes from the geometrical perspective are represented.The minimum distance and weight property lemmas and the theorem are proved for the generator matrix.Based upon the property of generator matrix,the structured algorithms of linear block codes are demonstrated.Since the complexity of optimal structured algorithm is very high,the binary linear block codes is searched by using the suboptimal structured algorithm.The comparison with Bose-Chaudhuri-Hocquenqhem（BCH） codes shows that the searched linear block codes are equivalent on minimum distance and can be designed for more block lengths.Because the linear block codes are used widely in communication systems and digital applications,the optimal and suboptimal structured algorithms must have great future being widely used in many applications and perspectives.

A NEW CLASS OF LINEAR CODES WITH TWO-LEVEL STRUCTURE

The binary extended Golay code has a two-level structure, which can be used in the decoding of the code. However, such structure is not limited to the Golay code, in fact, several binary linear codes can be constructed by a projective method which is related to the structure. In this correspondence, the binary (4n,n + 2k, ≥min(8, n,2d)) linear codes are resulted from quaternary (n,k,d) linear block codes. Based on the structure, the efficient maximum likelihood decoding algorithms can be presented correspondingly for the derived codes.

THE APPLICATION OF LINEAR COMPLEXITY OF SEQUENCES TO LOWER BOUNDS ON THE MINIMUM DISTANCE OF CYCLIC CODES

Firstly,the Fourier transforms in finite fields and the concept of linear complexityof sequences are described.Then several known lower bounds on the minimum distance of cycliccodes are outlined.Finally,the minimum distance of cyclic codes is analyzed via linear complexityof sequences,and new theorems about the lower bounds are obtained.

A Note on m-Weights of Linear Codes

m-weight, as a new generalization of classical Hamming weight, was discussedin this paper. A condition for the existence of linear codes of certain m-weights was given; theSingleton bound, Plotkin bound and Sphere Parking bound of Hamming weight were correspondinglygeneralized to the m-weight.

Linear Dispersion Orthogonal Space-Time Block Codes

A new architecture of space-time codes as a combination of orthogonal space-time block codes （OSTBC） and linear dispersion codes （LDC） is proposed in order to improve the bit error rate（BER） performance of OSTBC.The scheme proposed is named linear dispersion orthogonal space-time block codes （LDOSTBC）.In LDOSTBC scheme,firstly,the data is coded into LDC codewords.Then,the coded LDC substreams are coded into OSTBC codewords again.The decoding algorithm of LDOSTBC combines linear decoding of OSTBC and ML decoding or suboptimum detection algorithms of LDC.Compared with OSTBC scheme when the rate of LDC is MtR,the performance of LDOSTBC scheme can be improved without decreasing the data rate,where Mt is the number of transmit antennas and R is the spectral efficiency of the modulation constellation.If some rate penalty is allowed,when the rate of LDC is less than MtR the performance of LDOSTBC can be improved further.

Distinction of self-synchronous scrambled linear blockcodes based on multi-fractal spectrum

This study proposes a novel multi-fractal spectrumbasedapproach to distinguish linear block codes from its selfsynchronousscrambled codes. Given that the linear block codeand self-synchronous scrambled linear block code share the propertyof linear correlation, the existing linear correlation-basedidentification method is invalid for this case. This drawback can becircumvented by introducing a novel multi-fractal spectrum-basedmethod. Simulation results show that the new method has highrobustness and under the same conditions of bit error, the lowerthe code rate, the higher the recognition rate. Thus, the methodhas significant potential for future application in engineering.

NEW SECRET SHARING SCHEME BASED ON LINEAR CODE

A secret sharing system can be damaged when the dealer cheating occurs.In this paper,two kinds of secret sharing schemes based on linear code are proposed.One is a verifiable scheme which each participant can verify his own share from dealer's distribution and ensure each participant to receive valid share.Another does not have a trusted center,here,each participant plays a dual-role as the dealer and shadow(or share) provider in the whole scheme.

THE AUTOMORPHISM GROUP OF A LINEAR CODE

We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-

A Novel Decoder Based on Parallel Genetic Algorithms for Linear Block Codes

Genetic algorithms offer very good performances for solving large optimization problems, especially in the domain of error-correcting codes. However, they have a major drawback related to the time complexity and memory occupation when running on a uniprocessor computer. This paper proposes a parallel decoder for linear block codes, using parallel genetic algorithms (PGA). The good performance and time complexity are confirmed by theoretical study and by simulations on BCH(63,30,14) codes over both AWGN and flat Rayleigh fading channels. The simulation results show that the coding gain between parallel and single genetic algorithm is about 0.7 dB at BER = 10﹣5 with only 4 processors.

End-to-end verifiable electronic voting scheme of blockchain based on random linear block code

Blockchain is an emerging decentralized technology of electronic voting.The current main consensus protocols are not flexible enough to manage the distributed blockchain nodes to achieve high efficiency of consensus.For practical implementation,the consensus based on random linear block code(RLBC)is proposed and applied to blockchain voting scheme.Along with achieving the record correctness and consistency among all nodes,the consensus method indicates the active and inactive consensus nodes.This ability can assist the management of consensus nodes and restrain the generating of chain forks.To achieve end-to-end verifiability,cast-or-audit and randomized partial checking(RPC)are used in the proposed scheme.The voter can verify the high probability of correctness in ballot encryption and decryption.The experiments illustrate that the efficiency of proposed consensus is suitable for blockchain.The proposed electronic voting scheme is adapted to practical implementation of voting.

On the Computing of the Minimum Distance of Linear Block Codes by Heuristic Methods

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.

A Novel Low-bit-rate Speech Coding Based on Decomposition of the Pitch-cycle Waveform of the Linear Predictive Residual

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Processing for Aberrant Sampled Intensities in Linearly Coded Profilometry

Aiming at the local intensity anamorphosis in general linearly coded profilometry, the post treatment methods under different decoding conditions are proposed, in which the distorted intensities are rehabilitated or eliminated by our routines. The experimental results show that the measuring error and system cost can be effectively reduced while the reliability can be improved by introducing those methods into decoding process.