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- Hadamard_code abstract "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.A famous application of the Hadamard code was the NASA space probe Mariner 9 in 1971, where the code was used to transmit photos of Mars back to Earth.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 and also known under the names Walsh code or Walsh family and Walsh–Hadamard code, 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 Hadamard code is the same as the first order Reed–Muller code over the binary alphabet.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 2n 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 less 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.".
- Hadamard_code thumbnail Hadamard-Code.svg?width=300.
- Hadamard_code wikiPageExternalLink ld-binary-ms.pdf.
- Hadamard_code wikiPageExternalLink complexity.
- Hadamard_code wikiPageExternalLink lect4.pdf.
- Hadamard_code wikiPageID "7428961".
- Hadamard_code wikiPageRevisionID "567358735".
- Hadamard_code hasPhotoCollection Hadamard_code.
- Hadamard_code name "Hadamard code".
- Hadamard_code name "Punctured Hadamard code".
- Hadamard_code namesake Jacques_Hadamard.
- Hadamard_code notation "-code".
- Hadamard_code type Block_code.
- Hadamard_code type Linear_code.
- Hadamard_code subject Category:Coding_theory.
- Hadamard_code subject Category:Error_detection_and_correction.
- Hadamard_code type Abstraction100002137.
- Hadamard_code type Code106667317.
- Hadamard_code type Communication100033020.
- Hadamard_code type WrittenCommunication106349220.
- Hadamard_code comment "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.A famous application of the Hadamard code was the NASA space probe Mariner 9 in 1971, where the code was used to transmit photos of Mars back to Earth.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 and also known under the names Walsh code or Walsh family and Walsh–Hadamard code, 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 Hadamard code is the same as the first order Reed–Muller code over the binary alphabet.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. ".
- Hadamard_code label "Hadamard code".
- Hadamard_code label "Hadamard-Code".
- Hadamard_code label "アダマール符号".
- Hadamard_code sameAs Hadamard-Code.
- Hadamard_code sameAs アダマール符号.
- Hadamard_code sameAs m.0261bhc.
- Hadamard_code sameAs Q1567339.
- Hadamard_code sameAs Q1567339.
- Hadamard_code sameAs Hadamard_code.
- Hadamard_code wasDerivedFrom Hadamard_code?oldid=567358735.
- Hadamard_code depiction Hadamard-Code.svg.
- Hadamard_code isPrimaryTopicOf Hadamard_code.