Showing papers by "Irving S. Reed published in 1978"
TL;DR: It is shown that Reed-Solomon codes can be decoded by using a fast Fourier transform (FFT) algorithm over finite fields GF(F_{n}) , where F_{n} is a Fermat prime, and continued fractions.
Abstract: It is shown that Reed-Solomon (RS) codes can be decoded by using a fast Fourier transform (FFT) algorithm over finite fields GF(F_{n}) , where F_{n} is a Fermat prime, and continued fractions. This new transform decoding method is simpler than the standard method for RS codes. The computing time of this new decoding algorithm in software can be faster than the standard decoding method for RS codes.
41 citations
TL;DR: It is shown that \sqrt\[8]{2} is an element of order 2^{n+4} in GF(F_{n}) , where F_{n}=2^{2^{n}}+1 is a Fermat prime for n=3,4 .
Abstract: It is shown that \sqrt\[8]{2} is an element of order 2^{n+4} in GF(F_{n}) , where F_{n}=2^{2^{n}}+1 is a Fermat prime for n=3,4 . Hence it can be used to define a fast Fourier transform (FFT) of as many as 2^{n+4} symbols in GF(F_{n}) . Since \sqrt[8]{2} is a root of unity of order 2^{n+4} in GF(F_{n}) , this transform requires fewer muitiplications than the conventional FFT algorithm. Moreover, as Justesen points out [1], such an FFT can be used to decode certain Reed-Solomon codes. An example of such a transform decoder for the case n=2 , where \sqrt{2} is in GF(F_{2})=GF(17) , is given.
23 citations
TL;DR: In this paper, it was shown that the fast digital convolution technique used for parallel x-ray beams can be used also to reconstruct a density function for diverging xray beams, which when combined with what was called previously a generalized???-filter yields good accuracy, a flexibility to cope with noise and a substantial reduction in the xray dosage.
Abstract: It is shown here that the fast digital convolution technique used for parallel x-ray beams can be used also to reconstruct a density function for diverging x-ray beams. This technique when combined with what was called previously a generalized ???-filter yields good accuracy, a flexibility to cope with noise and a substantial reduction in the x-ray dosage. Finite field transforms and zero-order interpolation can be used also to improve the speed of the x-ray reconstruction process. Tests based on these methods are performed on an ideal phantom. A detailed comparison is made between a system using diverging beams, and a system using parallel beams with simulated data.
11 citations
01 May 1978
TL;DR: It is shown that Winograd's algorithm can be used to compute an integer transform over GF(q), where q is a Mersenne prime, which makes it possible to more easily encode b.h.c. and r.s. codes.
Abstract: It is shown that Winograd's algorithm can be used to compute an integer transform over GF(q), where q is a Mersenne prime. This new algorithm requires fewer multiplications than the conventional fast Fourier transform (f.f.t). The transform over GF(q) can be implemented readily on a digital computer. This fact makes it possible to more easily encode b.c.h. and r.s. codes.
9 citations
TL;DR: This new hybrid algorithm requires fewer multiplications than any previously known algorithm and is a combination of a Winograd algorithm and a fast complex integer transform developed previously by the authors.
Abstract: In this paper it is shown that the cyclic convolution of complex values can be performed by a hybrid transform. This transform is a combination of a Winograd algorithm and a fast complex integer transform developed previously by the authors. This new hybrid algorithm requires fewer multiplications than any previously known algorithm.
9 citations
TL;DR: For certain large transform lengths, Winograd's algorithm for computing the discrete Fourier transform (d.f.t.) is extended considerably by performing the cyclic convolution, required by Winog Rad's method, by a fast transform over certain complex integer fields developed previously by the authors.
Abstract: For certain large transform lengths, Winograd's algorithm for computing the discrete Fourier transform (d.f.t.) is extended considerably. This is accomplished by performing the cyclic convolution, required by Winograd's method, by a fast transform over certain complex integer fields developed previously by the authors. This new algorithm requires fewer multiplications than either the standard fast Fourier transform (f.f.t.) or Winograd's more conventional algorithm.
6 citations
TL;DR: A simple method is developed for computing elements of order 2kn, where n|2p?1?1 and 2 ?
Abstract: A simple method is developed for computing elements of order 2kn, where n|2p?1?1 and 2 ? k ? p+1, in the Galois field GF(q2), and q = 2p?1 is a Mersenne prime. Such primitive elements are needed to implement complex number-theoretic transforms.
2 citations
TL;DR: A Quick method is developed to find an element or order 2p+1p in the finite field GF(q2), where q = 2P−1 is a Mersenne prime, needed to implement complex integer transforms of length 2kp over GF( q2 where 3 ≤ k ≤ p + 1.
Abstract: A Quick method is developed to find an element or order 2p+1p in the finite field GF(q2), where q = 2P−1 is a Mersenne prime. Such an element is needed to implement complex integer transforms of length 2kp over GF(q2 where 3 ≤ k ≤ p + 1.
2 citations
2 citations
TL;DR: A fast algorithm is developed to calculate syndromes of multichannel linear systematic codes, including both block and convolutional codes, by using a direct sum of Galois fields.
Abstract: Classes of codes for a multichannel communication system are considered. A fast algorithm is developed to calculate syndromes of multichannel linear systematic codes, including both block and convolutional codes, by using a direct sum of Galois fields.
1 citations