Shigeichi Hirasawa

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.

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

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. >

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.

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

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.

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.

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.

Certain generalizations of concatenated codes--Exponential error bounds and decoding complexity

New coding and decoding schemes based on concatenated codes are proposed. The new coding schemes are potentially superior to Forneys original concatenation scheme in the sense that for discrete memoryless channels the former has a smaller upper bound on the probability of decoding error for the same order of decoding complexity.

On predictive errors of SQL injection attack detection by the feature of the single character

The sigmoid function has been widely used in the problem of the two value distinction. In this paper, we prepare a function which is similar to a sigmoid function, and show that our proposed model is valid to the SQL Injection attack detection than the method using a sigmoid function.

On automatic detection of SQL injection attacks by the feature extraction of the single character

The SQL injection attack causes very serious problem to web applications which have database including personal data. To detect the SQL injection attack, the parsing and the black list based on the existed attack have been widely used. However, a new attack method that is not included by the black list has still been developed. In this paper, we proposed the detecting method based on single character, and show the effectiveness of the proposed method experimentally using both attack and normal samples.