FAST ALGORITHM FOR REED-SOLOMON CODE AND THE HARDWARE IMPLEMENTATION

This paper described a signal processor for the Reed-Solomon (R-S) code using micro-programming. For the purpose of fast encoding and decoding,a formula for detecting two errors is derived, and the Qian search method for the decoding process is improved. The number of searches is significantly reduced from 256 to 4. At the same time, the circuit is simplified and the speed is increased. For the convenience of programming, a micro-programming compiling package is developed. The package can be used for programming of different formats of R-S code signal processors of DAB, MD and DCC. The software and the hardware can be used for error correcting, error detecting and error compensation of different formats of R-S code.

On Reed-Solomon Codes

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.

Incorporating red blanket protocol within code crimson:Streamlining definitive trauma care amid the chaos

The care of a patient involved in major trauma with exsanguinating haemorrhage is time-critical to achieve definitive haemorrhage control,and it requires coordinated multidisciplinary care.During initial resuscitation of a patient in the emergency department(ED),Code Crimson activation facilitates rapid decisionmaking by multi-disciplinary specialists for definitive haemorrhage control in operating theatre(OT)and/or interventional radiology(IR)suite.Once this decision has been made,there may still be various factors that lead to delay in transporting the patient from ED to OT/IR.Red Blanket protocol identifies and addresses these factors and processes which cause delay,and aims to facilitate rapid and safe transport of the haemodynamically unstable patient from ED to OT,while minimizing delay in resuscitation during the transfer.The two processes,Code Crimson and Red Blanket,complement each other.It would be ideal to merge the two processes into a single protocol rather than having two separate workflows.Introducing these quality improvement strategies and coor-dinated processes within the trauma framework of the hospitals/healthcare systems will help in further improving the multi-disciplinary care for the complex trauma patients requiring rapid and definitive haemorrhage control.

一套实用的企业应用分析框架CODE

随着企业信息化的深入,众多应用不再是“孤岛”,而是形成体系,应用间集成度越来越高,与业务的融合不断加深。在这种大趋势下,企业信息化在具体应用落地时要充分考虑其在企业体系中的定位,基于业务和技术环境找准发力点,才能最大化应用的价值。提出一套用于企业应用分析框架CODE,该分析框架由目标、业务上下文、系统上下文、领域、数据、分布、发展、接口、交互等要素构成,并充分考虑了企业环境的复杂性。经验证,该应用框架能够较好地完成应用分析,助力企业更好地进行应用建设。

Design of Serially Concatenated Two-Level Polar Coded Modulation System with Low-Complexity

Multilevel coding(MLC)is a commonly used polar coded modulation scheme,but challenging to implement in engineering due to its high complexity and long decoding delay for high-order modulations.To address these limitations,a novel two-level serially concatenated MLC scheme,in which the bitlevels with similar reliability are bundled and transmitted together,is proposed.The proposed scheme hierarchically protects the two bit-level sets:the bitlevel sets at the higher level are sufficiently reliable and do not require excessive resources for protection,whereas only the bit-level sets at the lower level are encoded by polar codes.The proposed scheme has the advantages of low power consumption,low delay and high reliability.Moreover,an optimized constellation signal labeling rule that can enhance the performance is proposed.Finally,the superiority of the proposed scheme is validated through the theoretical analysis and simulation results.Compared with the bit interleaving coding modulation(BICM)scheme,under 256-quadrature amplitude modulation(QAM),the proposed scheme attains a performance gain of 1.0 dB while reducing the decoding complexity by 54.55%.

Low Complexity Successive Cancellation List Decoding of U-UV Codes

Constituted by BCH component codes and its ordered statistics decoding(OSD),the successive cancellation list(SCL)decoding of U-UV structural codes can provide competent error-correction performance in the short-to-medium length regime.However,this list decoding complexity becomes formidable as the decoding output list size increases.This is primarily incurred by the OSD.Addressing this challenge,this paper proposes the low complexity SCL decoding through reducing the complexity of component code decoding,and pruning the redundant SCL decoding paths.For the former,an efficient skipping rule is introduced for the OSD so that the higher order decoding can be skipped when they are not possible to provide a more likely codeword candidate.It is further extended to the OSD variant,the box-andmatch algorithm(BMA),in facilitating the component code decoding.Moreover,through estimating the correlation distance lower bounds(CDLBs)of the component code decoding outputs,a path pruning(PP)-SCL decoding is proposed to further facilitate the decoding of U-UV codes.In particular,its integration with the improved OSD and BMA is discussed.Simulation results show that significant complexity reduction can be achieved.Consequently,the U-UV codes can outperform the cyclic redundancy check(CRC)-polar codes with a similar decoding complexity.

Hybrid DF and SIR Forwarding Strategy in Conventional and Distributed Alamouti Space-Time Coded Cooperative Networks

In this paper,we propose a hybrid decode-and-forward and soft information relaying(HDFSIR)strategy to mitigate error propagation in coded cooperative communications.In the HDFSIR approach,the relay operates in decode-and-forward(DF)mode when it successfully decodes the received message;otherwise,it switches to soft information relaying(SIR)mode.The benefits of the DF and SIR forwarding strategies are combined to achieve better performance than deploying the DF or SIR strategy alone.Closed-form expressions for the outage probability and symbol error rate(SER)are derived for coded cooperative communication with HDFSIR and energy-harvesting relays.Additionally,we introduce a novel normalized log-likelihood-ratio based soft estimation symbol(NL-SES)mapping technique,which enhances soft symbol accuracy for higher-order modulation,and propose a model characterizing the relationship between the estimated complex soft symbol and the actual high-order modulated symbol.Further-more,the hybrid DF-SIR strategy is extended to a distributed Alamouti space-time-coded cooperative network.To evaluate the~performance of the proposed HDFSIR strategy,we implement extensive Monte Carlo simulations under varying channel conditions.Results demonstrate significant improvements with the hybrid technique outperforming individual DF and SIR strategies in both conventional and distributed Alamouti space-time coded cooperative networks.Moreover,at a SER of 10^(-3),the proposed NL-SES mapping demonstrated a 3.5 dB performance gain over the conventional averaging one,highlighting its superior accuracy in estimating soft symbols for quadrature phase-shift keying modulation.

On error distance of Reed-Solomon codes

The complexity of decoding the standard Reed-Solomon code is a well known open prob-lem in coding theory. The main problem is to compute the error distance of a received word. Using the Weil bound for character sum estimate, we show that the error distance can be determined precisely when the degree of the received word is small. As an application of our method, we give a significant improvement of the recent bound of Cheng-Murray on non-existence of deep holes (words with maximal error distance).

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes [p-1, k]p with p a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes [q-1, k]q with q a power of the prime p. Let q≥4 and 2≤k≤q-2. We show that the received word u is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree q-2 and a polynomial of degree at most k-1. So there are at least 2(q-1)qk deep holes if k q-3.

A Low Power Error Detection in the Syndrome Calculator Block for Reed-Solomon Codes: RS(204,188)

Reed-Solomon （RS） codes have been widely adopted in many modern communication systems. This paper describes a new method for error detection in the syndrome calculator block of RS decoders. The main feature of this method is to prove that it is possible to compute only a few syndrome coeffi- cients -- less than half-- to detect whether the codeword is correct. The theoretical estimate of the prob- ability that the new algorithm failed is shown to depend on the number of syndrome coefficients computed. The algorithm is tested using the RS（204,188） code with the first four coefficients. With a bit error rate of 1 ~ 104, this method reduces the power consumption by 6% compared to the basic RS（204,188） decoder. The error detection algorithm for the syndrome calculator block does not require modification of the basic hardware implementation of the syndrome coefficients computation. The algorithm significantly reduces the computation complexity of the syndrome calculator block, thus lowering the power needed.

New concatenated soft decoding of Reed-Solomon codes with lower complexities

To improve error-correcting performance, an iterative concatenated soft decoding algorithm for Reed-Solomon （RS） codes is presented in this article. This algorithm brings both complexity as well as advantages in performance over presently popular sot~ decoding algorithms. The proposed algorithm consists of two powerful soft decoding techniques, adaptive belief propagation （ABP） and box and match algorithm （BMA）, which are serially concatenated by the accumulated log-likelihood ratio （ALLR）. Simulation results show that, compared with ABP and ABP-BMA algorithms, the proposed algorithm can bring more decoding gains and a better tradeoff between the decoding performance and complexity.

High Speed Versatile Reed-Solomon Decoder for Correcting Errors and Erasures

A new Chien search method for shortened Reed-Solomon （RS） code is proposed, based on this, a versatile RS decoder for correcting both errors and erasures is designed. Compared with the traditional RS decoder, the weighted coefficient of the Chien search method is calculated sequentially through the three pipelined stages of the decoder. And therefore, the computation of the errata locator polynomial and errata evaluator polynomial needs to be modified. The versatile RS decoder with minimum distance 21 has been synthesized in the Xilinx Virtex-Ⅱ series field programmable gate array （FPGA） xe2v1000-5 and is used by coneatenated coding system for satellite communication. Results show that the maximum data processing rate can be up to 1.3 Gbit/s.

On Deep Holes of Projective Reed-Solomon Codes over Finite Fields with Even Characteristic

Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.

On Ordinary Words of Standard Reed-Solomon Codes over Finite Fields

Reed-Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm.Usually we use the maximum likelihood decoding(MLD)algorithm in the decoding process of Reed-Solomon codes.MLD algorithm relies on determining the error distance of received word.Dür,Guruswami,Wan,Li,Hong,Wu,Yue and Zhu et al.got some results on the error distance.For the Reed-Solomon code C,the received word u is called an ordinary word of C if the error distance d(u,C)=n-deg u(x)with u(x)being the Lagrange interpolation polynomial of u.We introduce a new method of studying the ordinary words.In fact,we make use of the result obtained by Y.C.Xu and S.F.Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed-Solomon codes over the finite field of q elements.This completely answers an open problem raised by Li and Wan in[On the subset sum problem over finite fields,Finite Fields Appl.14(2008)911-929].

Finite Geometry and Deep Holes of Reed-Solomon Codes over Finite Local Rings

In this paper, we first propose the maximum arc problem, normal rational curve conjecture, and extensions of normal rational curves over finite local rings, analogously to the finite geometry over finite fields. We then study the deep hole problem of generalized Reed-Solomon (RS) codes over finite local rings. Several different classes of deep holes are constructed. The relationship between finite geometry and deep holes of RS codes over finite local rings are also studied.

SUBSPACE SUBCODES OF GENERALIZED REED-SOLOMON CODES

In this paper, the subspace subcodes of generalized Reed-Solomn codes are codes are introduced and the fomulas to compute the dimensions of these codes are given.

Homogeneous Interpolation Problem and Key Equation for Decoding Reed-Solomon Codes

The concept of homogeneous interpolation problem (HIP) over fields is introduced.It is discovered that solving HIP over finite fields is equivalent to decoding Reed-Solomon (RS) codes.The Welch-Berlekamp algorithm of decoding RS codes is derived;besides,by introducing the concept of incomplete locator of error patterns,the algorithm called incomplete iterative decoding is established.

A Theoretical Method for Estimating Performance of Reed-Solomon Codes Concatenated with Orthogonal Space-Time Block Codes

Based on the studies of Reed-Solomon codes and orthogonalspace-time block codes over Rayleigh fading channel, a theoreticalmethod for estimating performance of Reed-Solomon codes concatenatedwith orthogonal space- time block codes is presented in this paper.And an upper bound of the bit error rate is also obtained. It isshown through computer simulations that the signal-to-noise ratioreduces about 15 dB or more after orthogonal space-time block codesare concatenate with Reed-Solomon(15,6)codes over Rayleigh fadingchannel, when the bit error rate is 10^-4.

ENUMERATION RESULTS FOR THE CODEWORDS HAVING NO INNER PERIODS IN REED-SOLOMON CODES

Let x=(x0, …, xn-1) be a sequence in the finite field GF（q） with length n, Si, x is the i-cyclic shift of x,i.e. Six=(xi, xi+1, …, xi-1) （where i+1 means （i+1）rood n）. If there exists a positive integer 0【r≤n making Srx=x+（u, u, …, u） hold for some u∈GF（q）, then the r is called one of the generalized periods of this sequence x. The least one rmin of such periods is called the minimum generalized period of x. In narticular, if rmin=n (i. e. the

Iterative list decoding approach for Reed-Solomon codes

A novel adaptively iterative list decoding(ILD) approach using for Reed-Solomon(RS) codes was investigated. The proposed scheme is exploited to reduce the complexity of RS Chase algorithm(CA) via an iterative decoding attempt mode. In each decoding attempt process, a test pattern is generated by flipping the bits of the least reliable positions(LRPs) within the received hard-decision(HD) vector. The ILD algorithm continues until a test pattern is successfully decoded by the underlying Berlekamp-Massey algorithm(BMA) of RS codes. Flipping within the same bits, the ILD algorithm provides the same test pattern set as the conventional RS CA, thus there is no degradation in error-rate performance. Without decoding all test patterns, the ILD algorithm can simplify the decoding complexity by its early termination. Simulation results show that the average complexity of the ILD algorithm is much lower than that of the conventional RS CA(and is similar to that of BMA decoding) at high signal-to-noise ratio(SNR) region with no less to the RS CA decoding error-rate performance.