Decoding of Reed-Solomon codes for additive cost functions

Non-binary multithreshold decoding (qMTD) for q-ary self-orthogonal codes (qSOC) is considered. The SER performance of qMTD is shown to be close to the results provided by optimum total search methods, which are not realizable for non-binary codes in general. qMTD decoders are compared with different decoders for Reed-Solomon codes. The performance provided with qMTD in some cases is unattainable with classical decoders for arbitrary long Reed-Solomon codes. The result of concatenation of qSOC with simple to decode outer codes is described. Method of improving of qMTD decoder's performance for qSOC is offered. Some simulated results obtained by using these two decoding techniques (the base and modified ways) are presented as well. Comparison of the results showed that the change in the threshold element's algorithm can significantly improve speed of qMTD work. It's shown that for a larger gain this modification qMTD should be used after conventional decoding iterations.

In this chapter, we will give information about Reed-Solomon codes. These codes fall into the category of nonbinary cyclic codes. The generator polynomials of Reed-Solomon codes are constructed using the minimal polynomials of the extended finite fields. Reed-Solomon codes are effective for burst errors and they are used for erasure decoding. Reed-Solomon codes are used in some electronic devices such as CDs, DVDs, and Blu-ray, and they are also employed in communication technologies such as DSL, WiMAX, or RAID 6. Reed-Solomon codes are invented in 1960, and they are seen as nonbinary BCH codes. In this chapter, we first explain the construction of the generator polynomials of the Reed-Solomon codes using the minimal polynomials of the extended finite fields. Next, we provide information about the syndrome decoding of Reed-Solomon codes using error evaluator polynomial. In sequel, Berlekamp algorithm which is a low-complexity syndrome decoding algorithm used for the decoding of Reed-Solomon codes is explained.

Algebraic soft-decision decoding (ASD) of Reed-Solomon (RS) codes can provide substantial coding gain with polynomial complexity. Among the ASD algorithms with practical multiplicity assignment schemes, the Low-Complexity Chase (LCC) decoding can achieve similar or higher coding gain. Interpolation is a major step in ASD. Since the maximum multiplicity of the interpolation point is only one in LCC, the interpolation over each point has low complexity. However, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors are involved in the LCC, and the interpolation needs to be carried out on each of them. In order to reduce the computational complexity of the overall interpolation, intermediate results can be stored and shared. Nevertheless, the storage requires large memory, which accounts for a significant portion of the overall hardware requirement of the interpolator. In this paper, we propose a novel interpolation procedure, in which the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors are mapped to the vertices of a dimension-η hypercube and the vectors mapped to adjacent vertices have only one different entry. In addition, a backward interpolation is proposed to support the traversal from one vertex to its neighbors. Traveling through the entire hypercube, the interpolation over each test vector can be done one after another and the memory requirement is reduced by a factor of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η-1</sup> . Efficient architectures are also developed for the proposed interpolation procedure. With about the same latency and the same number of gates as in prior efforts, our architecture can reduce the memory size to 25% and the number of registers to 57% for a (255, 239) RS code with η = 3. The saving further increases with η.

A formula for the determination of error values for a Hermitian code is given. The formula can be seen as a generalization of Forneys' formula that applies in the decoding of Reed-Solomon codes.

The main goal is to reveal the 1-concavity property for a subclass of cost games called data cost games. The motivation for the study of the 1-concavity property is the appealing theoretical results for both the core and the nucleolus, in particular their geometrical characterization as well as their additivity property. The characteristic cost function of the original data cost game assigns to every coalition the additive cost of reproducing the data the coalition does not own. The underlying data and cost sharing situation is composed of three components, namely, the player set, the collection of data sets for individuals, and the additive cost function on the whole data set. The proof of 1-concavity is direct, but robust to a suitable generalization of the characteristic cost function. As an adjunct, the 1-concavity property is shown for the subclass of so-called “bicycle” cost games, inclusive of the data cost games in which the individual data sets are nested in a decreasing order.

The paper presents a systematic approach for finding the optimal set of predictors for consistent identification of a single transfer function in an acyclic dynamic network. It is assumed that the topology of the network is known, the forcing inputs are not measured, and the observations have positive additive costs. For a class of networks where the target node is not involved in a feedback loop, sufficient and necessary conditions are derived to consistently identify a certain transfer function via a multi-input single-output prediction error method. This enables designing a systematic graphical approach based on the notion of d-separation to look for an optimal set of predictors that minimizes an appropriate additive cost function. It is shown that the required conditions for consistency and optimality are equivalent to the notion of separation in an undirected graph resulted from systematically manipulating the graphical representation of the network. Then, some well-known algorithms from computer science can be used to find the optimal set of predictors.

Coupled tensor factorization methods are useful for sensor fusion, combining information from several related datasets by simultaneously approximating them by products of latent tensors. In these methods, the choice of a suitable optimization criteria becomes difficult as observed datasets may have different statistical characteristics and their relative importance for the task at hand can vary. In this paper, we present an algorithmic framework for coupled factorization that, while estimating a latent factorization also estimates a specific ß-divergence for each dataset as well as the relative weights in an overall additive cost function. We evaluate the proposed method on both synthetical and real datasets, where we apply our methods on a link prediction problem. The results show that our method outperforms the state-of-the-art by a significant margin.

Differential dissipativity based distributed MPC for flexible operation of nonlinear plantwide systems

In this paper, we present a new fast maximum likelihood (ML) detection algorithm for quasi-orthogonal space-time block codes ( QO-STBC). Observed in a transformed signal space, the QO-STBC encoder embeds information into a group of orthogonal subspaces of rank two. We propose a fast technique that separates information embedded in each rank-two subspace by a simple linear processing, and decodes the separated information independently. For a multiple-input and multiple-output (MIMO) system, decoding algorithm decomposes the original N /spl times/ N system into N/2 parallel 2 /spl times/ 2 MIMO sub-systems without information loss followed by MRC. It then detects symbols contained in each 2 /spl times/ 2 MIMO sub-system using a sphere decoder of reduced computational complexity.

We investigate the translation equation (T) , and the cocycle equations (Co1) , (Co2) in [ x ], the ring of formal power series over . Here we restrict ourselves to iteration groups of type I, i. e. to solutions of (T) of the form , where c 1 ? 1 is a generalized exponential function. It is easy to prove that the coefficient functions c n (s), a n (s), s n (s) of are polynomials, c 1(s), polynomials or rational functions in one more generalized exponential function and, in a special case, polynomials in an additive function. For example, we obtain . For this we do not need any detailed information about the polynomials P n . Under some conditions on the exponential and additive functions it is possible to replace the exponential and additive functions by independent indeterminates. In this way we obtain formal versions of the translation equation and the cocycle equations in rings of the form [ x ], [ x ], [ x ], [ x ], and [ x ]. We solve these equations in a completely algebraic way, by deriving formal differential equations or Aczel�Jabotinsky type equations for these problems. It is possible to get the structure of the coefficients in great detail which are now polynomials or rational functions. We prove the universal character (depending on certain parameters) of these polynomials or rational functions. And we deduce the canonical form S -1(yS(x)) for iteration groups of type I. This approach seems to be more clear and more general than the original one. Some simple substitutions allow us to solve these problems in rings of the form ( [ u ]) [ x ], i. e. where the coefficient functions are formal series.

In this paper, we consider the design of the day-ahead market for the smart electrical grid. Consumers with flexible demand and generator companies participate in the market to settle on their load and supply schedules, respectively. The market is operated by an Independent System Operator (ISO) whose purpose is to maximize social welfare while keeping load and supply balanced in the electricity network. We develop two distributed pricing algorithms that achieve optimum welfare. The first algorithm yields time-dependent market prices under convexity assumptions on utility and cost functions and the second algorithm yields bundle prices for arbitrary utility and cost functions. In both algorithms, flexible consumers and generator companies simply determine their own schedules based on the prices updated by the ISO at each iteration. We show that the participation of flexible demand in the day-ahead market reduces supply volatility, which would be present when flexible demand does not take part in price setting procedure.

As gate delays decrease faster than wire delays for each technology generation, buffer insertion becomes a popular method to reduce the interconnect delay. Several modern buffer insertion algorithms (e.g., [7, 6, 15]) are based on van Ginneken's dynamic programming paradigm [14]. However, van Ginneken's original algorithm does not control buffering resources and tends to over-buffering, thereby wasting area and power. It has been a major open problem whether it is possible to optimize slack and at the same time minimize the buffer usage.This paper settles this open problem by showing that for arbitrary integer cost functions, the problem is NP-complete. We also extend the pre-buffer slack technique [12] to minimize the buffer cost. This technique can significantly reduce the running time and memory in buffer cost minimization problem. The experimental results show that our algorithm can speed up the running time up to 17 times and reduces the memory to 1/30 of traditional best know algorithm. Finally, we show how to efficiently deal with multiway merge in buffer insertion.

The authors consider expansions which give arbitrary orthonormal tilings of the time-frequency plane. These differ from the short-time Fourier transform, wavelet transform, and wavelet packets tilings in that they change over time. They show how this can be achieved using time-varying orthogonal tree structures, which preserve orthogonality, even across transitions. The method is based on the construction of boundary and transition filters; these allow us to construct essentially arbitrary tilings. Time-varying modulated lapped transforms are a special case, where both boundary and overlapping solutions are possible with filters obtained by modulation. They present a double-tree algorithm which for a given signal decides on the best binary segmentation in both time and frequency. That is, it is a joint optimization of time and frequency splitting. The algorithm is optimal for additive cost functions (e.g., rate-distortion), and results in time-varying best bases, the main application of which is for compression of nonstationary signals. Experiments on test signals are presented.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The problem of efficient maximum-likelihood soft decision decoding of binary BCH codes is considered. It is known that those primitive BCH codes whose designed distance is one less than a power of two, contain subcodes of high dimension which consist of a direct-sum of several identical codes. The authors show that the same kind of direct-sum structure exists in all the primitive BCH codes, as well as in the BCH codes of composite block length. They also introduce a related structure termed the concurring-sum, and then establish its existence in the primitive binary BCH codes. Both structures are employed to upper bound the number of states in the minimal trellis of BCH codes, and develop efficient algorithms for maximum-likelihood soft decision decoding of these codes. >

We consider a general online scheduling problem on a single machine with the objective of minimizing Σjwjg(Fj), where wj is the weight/importance of job Jj, Fj is the flow time of the job in the schedule, and g is an arbitrary non-decreasing cost function. Numerous natural scheduling objectives are special cases of this general objective. We show that the scheduling algorithm Highest Density First (HDF) is (2+e)-speed O(1)-competitive for all cost functions g simultaneously. We give lower bounds that show the HDF algorithm and this analysis are essentially optimal. Finally, we show scalable algorithms are achievable in some special cases.