Toshihiko Namekawa

A method for solving key equation for decoding goppa codes

In this paper we show that the key equation for decoding Goppa codes can be solved using Euclids 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 Euclids 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.

Direct Evaluation of Radar Detection Probabilities

A simple and effective procedure for evaluating detectionperformances in radar and sonar detection problems is derived forboth fixed-threshold and adaptive-threshold detection. Using theprocedure, the cumulative probabilities of the test statistic can bedirectly evaluated from the moment generating functions bycalculating residues. The exact formulae for computing the detectionperformances for the chi-square family of fluctuating targets withan integer fluctuation parameter are given in a finite sum formwithout any special functions for both fixed threshold and cellaverageconstant false-alarm rate detection by using the methoddeveloped here.

Error Probability Characteristics for CPSK Signal Through m-Distributed Fading Channel

The average error probabilities and outage rates of error probability for the M -phase CPSK signal through the Nakagamis m -distributed fading channel are exactly evaluated both for nondiversity reception and for diversity reception. The probability density functions of the composite phase of fading signal and noise and those of the diversity-combined signal envelopes are newly derived in this paper. The results are generally obtained including the digital phase modulation component, the fading figure as a measure of fading depth, the average carrier-to-noise power ratio, the power correlation coefficient between two diversity branches, etc. The diversity improvements are also verified. Additionally some useful approximate formulas are briefly shown. This study will add a newly widened view to the considerations of system performances and designs for digital radio communications via fading channels.

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

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 Euclids algorithm.

Pulse Interval and Width Modulation for Video Transmission

This paper proposes new pulse analog modulation system; Pulse Interval and Width Modulation suitable for light emitting diodes. In this system pulses have information on both their width and interval between them. Pulse Interval and Width Modulation is characterized by the following compared with Pulse Interval Modulation. First: Pulse repetition frequency is reduced by one half. Second: Duty ratio is 50 % on the average. Third: More power is needed while carrier to noise ratio could be improved instead. The paper shows how it works and comparison with another pulse analog modulation such as Pulse Frequency Modulation or Pulse Interval Modulation. The paper also shows an experiment of twin channel transmission employing two light emitting diodes, two optical fiber transmission lines and two PIN photodiodes as an example of application of Pulse Interval and Width Modulation.

New classes of binary codes constructed on the basis of concatenated codes and product codes

We present new classes of binary codes that are constructed on the basis of concatenated codes and product codes. We discuss the random-error-correction capabilities of these codes. Some examples of the codes for the correction of random errors are given which have at least as many codewords as the best codes previously known (to the authors) with the same minimum distance and same number of check symbols. The burst-error-correction capabilities of the codes are also discussed. Several examples of the codes for the correction of both random errors and burst errors are given. A decoding algorithm for the codes is also described.

An erasures-and-errors decoding algorithm for Goppa codes (Corresp.)

An erasures-and-errors decoding algorithm for Goppa codes is presented. Given the Goppa polynomial and the modified syndrome polynomial, a modified key equation is solved using Euclids algorithm to determine the error locator polynomial and the errata evaluator polynomial.

Double Symbol Error Rates of M-ary DPSK in a Satellite-Aircraft Multipath Channel

Double symbol error rates of M -ary differentially coherent phase shift keying (DPSK) systems in a satellite-aircraft multipath channel are evaluated. A joint probability density function (pdf) of two consecutive differential phase errors in the DPSK outputs is newly derived. Numerically calculated results are shown in figures for binary, quaternary, and octonary DPSK systems. It is further shown that the analytical values agree well with the data observed by ATS-5 experimental tests.

Modified product codes

By modifying product codes, a new coding scheme and its decoding method are proposed. Compared to a product code A , the first stage code A_{1} of the new code A_{M} is constructed in the same way as that of the code A except that it has at least one subcode, while the second stage codes A_{2}^{(j)} of the code A_{M} are a set of codes With the same length and different rates. The new coding scheme has a smaller upper bound on the probability of decoding error than the original product coding scheme for any given nonzero rate less than the capacity of a binary symmetric channel. An example is given for which the rate is increased compared With the original product code, at a fixed probability of decoding error, for a relatively short code length.

A New Optical Communication System Using the Pulse Interval and Width Modulated Code

This paper proposes a new optical communication system using the Pulse Interval and Width Modulated Code. It is constructed as combinations of pulse width and interval. Then it is easy to understand that the transmission capacity of this Code is bigger than those of Pulse Width Modulation and/or Differencial Pulse Position Modulation.