|
The Hadamard code is an error-correcting code that is used for error detection and correction when transmitting messages over very noisy or unreliable channels. In 1971, the code was used to transmit photos of Mars back to Earth from the NASA space probe Mariner 9 Because of its unique mathematical properties, the Hadamard code is not only used by engineers, but also intensely studied in coding theory, mathematics, and theoretical computer science. The Hadamard code is named after the French mathematician Jacques Hadamard. It is also known under the names Walsh code, Walsh family,〔See, e.g., 〕 and Walsh–Hadamard code〔See, e.g., .〕 in recognition of the American mathematician Joseph Leonard Walsh. The Hadamard code is an example of a linear code over a binary alphabet that maps messages of length to codewords of length . It is unique in that each non-zero codeword has a Hamming weight of exactly , which implies that the distance of the code is also . In standard coding theory notation for block codes, the Hadamard code is a -code, that is, it is a linear code over a binary alphabet, has block length , message length (or dimension) , and minimum distance . The block length is very large compared to the message length, but on the other hand, errors can be corrected even in extremely noisy conditions. The punctured Hadamard code is a slightly improved version of the Hadamard code; it is a -code and thus has a slightly better rate while maintaining the relative distance of , and is thus preferred in practical applications. The punctured Hadamard code is the same as the first order Reed–Muller code over the binary alphabet.〔See, e.g., .〕 Normally, Hadamard codes are based on Sylvester's construction of Hadamard matrices, but the term “Hadamard code” is also used to refer to codes constructed from arbitrary Hadamard matrices, which are not necessarily of Sylvester type. In general, such a code is not linear. Such codes were first constructed by R. C. Bose and S. S. Shrikhande in 1959. If ''n'' is the size of the Hadamard matrix, the code has parameters , meaning it is a not-necessarily-linear binary code with 2''n'' codewords of block length ''n'' and minimal distance ''n''/2. The construction and decoding scheme described below apply for general ''n'', but the property of linearity and the identification with Reed–Muller codes require that ''n'' be a power of 2 and that the Hadamard matrix be equivalent to the matrix constructed by Sylvester's method. The Hadamard code is a locally decodable code, which provides a way to recover parts of the original message with high probability, while only looking at a small fraction of the received word. This gives rise to applications in computational complexity theory and particularly in the design of probabilistically checkable proofs. Since the relative distance of the Hadamard code is 1/2, normally one can only hope to recover from at most a 1/4 fraction of error. Using list decoding, however, it is possible to compute a short list of possible candidate messages as long as fewer than of the bits in the received word have been corrupted. In code division multiple access (CDMA) communication, the Hadamard code is referred to as Walsh Code, and is used to define individual communication channels. It is usual in the CDMA literature to refer to codewords as “codes”. Each user will use a different codeword, or “code”, to modulate their signal. Because Walsh codewords are mathematically orthogonal, a Walsh-encoded signal appears as random noise to a CDMA capable mobile terminal, unless that terminal uses the same codeword as the one used to encode the incoming signal.〔(【引用サイトリンク】title=CDMA Tutorial: Intuitive Guide to Principles of Communications )〕 ==History== ''Hadamard code'' is the name that is most commonly used for this code in the literature. However, in modern use these error correcting codes are referred to as Walsh–Hadamard codes. There is a reason for this: Jacques Hadamard did not invent the code himself, but he defined Hadamard matrices around 1893, long before the first error-correcting code, the Hamming code, was developed in the 1940s. The Hadamard code is based on Hadamard matrices, and while there are many different Hadamard matrices that could be used here, normally only Sylvester's construction of Hadamard matrices is used to obtain the codewords of the Hadamard code. James Joseph Sylvester developed his construction of Hadamard matrices in 1867, which actually predates Hadamard's work on Hadamard matrices. Hence the name ''Hadamard code'' is not undisputed and sometimes the code is called ''Walsh code'', honoring the American mathematician Joseph Leonard Walsh. A Hadamard code was used during the 1971 Mariner 9 mission to correct for picture transmission errors. The data words used during this mission were 6 bits long, which represented 64 grayscale values. Because of limitations of the quality of the alignment of the transmitter at the time (due to Doppler Tracking Loop issues) the maximum useful data length was about 30 bits. Instead of using a repetition code, a (6, 16 ) Hadamard code was used. Errors of up to 7 bits per word could be corrected using this scheme. Compared to a 5-repetition code, the error correcting properties of this Hadamard code are much better, yet its rate is comparable. The efficient decoding algorithm was an important factor in the decision to use this code. The circuitry used was called the "Green Machine". It employed the fast Fourier transform which can increase the decoding speed by a factor of three. Since the 1990s use of this code by space programs has more or less ceased, and the Deep Space Network does not support this error correction scheme for its dishes that are greater than 26 m. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hadamard code」の詳細全文を読む スポンサード リンク
|