There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

A method is described for constructing long error-correcting codes from one or more shorter error-correcting codes, referred to as subcodes, and a bipartite graph. A graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes. Lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes. Both the encoders and decoders proposed are shown to take advantage of the code's explicit decomposition into subcodes to decompose and simplify the associated computational processes. Bounds on the performance of two specific decoding algorithms are established, and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed. The proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver, e.g., reliability information, without greatly increasing the number of computations required. It is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst-error correction abilities. The construction principles

A recursive approach to low complexity codes

Information Theory and Reliable Communication

In this paper we show that the key equation for decoding Goppa codes can be solved using Euclid's algorithm. The division for computing the greatest common divisor of the Goppa polynomial g(z) of degree 2t and the syndrome polynomial is stopped when the degree of the remainder polynomial is less than or equal to t − 1. The error locator polynomial is proved the multiplier polynomial for the syndrome polynomial multiplied by an appropriate scalar factor. The error evaluator polynomial is proved the remainder polynomial multiplied by an appropriate scalar factor. We will show that the Euclid's algorithm can be modified to eliminate multiplicative inversion, and we will evaluate the complexity of the inversionless algorithm by the number of memories and the number of multiplications of elements in GF(qm). The complexity of the method for solving the key equation for decoding Goppa codes is a few times as much as that of the Berlekamp—Massey algorithm for BCH codes modified by Burton. However the method is straightforward and can be applied for solving the key equation for any Goppa polynomial.

A method for solving key equation for decoding goppa codes

A new soft decoding algorithm for linear block codes is proposed. The decoding algorithm works with any algebraic decoder and its performance is strictly the same as that of maximum-likelihood-decoding (MLD). Since our decoding algorithm generates sets of different candidate codewords corresponding to the received sequence, its decoding complexity depends on the received sequence. We compare our decoding algorithm with Chase (1972) algorithm 2 and the Tanaka-Kakigahara (1983) algorithm in which a similar method for generating candidate codewords is used. Computer simulation results indicate, for some signal-to-noise ratios (SNR), that our decoding algorithm requires less average complexity than those of the other two algorithms, but the performance of ours is always superior to those of the other two. >

An efficient maximum-likelihood-decoding algorithm for linear block codes with algebraic decoder

It is shown that Goppa codes with Goppa polynomial \{g(z)\}^{q} have the parameters: length n \leq q^{m} - s_{o} , number of check symbols n - k \leq m (q - 1) (\deg g) , and minimum distance d \geq q (\deg g) + 1 , where q is a prime power, m is an integer, g(z ) is an arbitrary polynomial over GF(q^{m}) , and so is the number of roots of g(z) which belong to GF(q^{m}) . It is also shown that all binary Goppa codes of length n \leq 2^{m} - s_{o} satisfy the relation n - k \leq m (d - 1)/2 . A new class of binary codes with n \leq 2^{ m} + ms _{0}, n - k \leq m (\deg g) + s_{0} , and d \leq 2(\deg g) + 1 is constructed, as well as another class of binary codes with slightly different parameters. Some of those codes are proved superior to the best codes previously known. Finally, a decoding algorithm is given for the codes constructed which uses Euclid's algorithm.

Further results on Goppa codes and their applications to constructing efficient binary codes

The problem of distortionless encoding when the parameters of the probabilistic model of a source are unknown is considered from a statistical decision theory point of view. A class of predictive and nonpredictive codes is proposed that are optimal within this framework. Specifically, it is shown that the codeword length of the proposed predictive code coincides with that of the proposed nonpredictive code for any source sequence. A bound for the redundancy for universal coding is given in terms of the supremum of the Bayes risk. If this supremum exists, then there exists a minimax code whose mean code length approaches it in the proposed class of codes, and the minimax code is given by the Bayes solution relative to the prior distribution of the source parameters that maximizes the Bayes risk. >

A class of distortionless codes designed by Bayes decision theory

A new soft-decision maximum-likelihood decoding algorithm is proposed, which generates a set of candidate codewords using hard-decision bounded-distance decoding. By improving the generating method of input vectors for the bounded-distance decoding due to Kaneko et al. (see ibid., vol.40, no.3, p.320-27, 1994), the decoding time complexity is reduced without degradation of the performance. The space complexity is dependent on the bounded-distance decoding.

Shigeichi Hirasawa

Papers

A method for solving key equation for decoding goppa codes

An efficient maximum-likelihood-decoding algorithm for linear block codes with algebraic decoder

Further results on Goppa codes and their applications to constructing efficient binary codes

A class of distortionless codes designed by Bayes decision theory

An improvement of soft-decision maximum-likelihood decoding algorithm using hard-decision bounded-distance decoding