Walsh code

Hadamard code
Named after	Jacques Hadamard
Classification
Type	Linear block code
Block length	${\displaystyle n=2^{k}}$
Message length	${\displaystyle k}$
Rate	${\displaystyle k/2^{k}}$
Distance	${\displaystyle d=2^{k-1}}$
Alphabet size	${\displaystyle 2}$
Notation	${\displaystyle [2^{k},k,2^{k-1}]_{2}}$-code

Punctured Hadamard code
Named after	Jacques Hadamard
Classification
Type	Linear block code
Block length	${\displaystyle n=2^{k}}$
Message length	${\displaystyle k+1}$
Rate	${\displaystyle (k+1)/2^{k}}$
Distance	${\displaystyle d=2^{k-1}}$
Alphabet size	${\displaystyle 2}$
Notation	${\displaystyle [2^{k},k+1,2^{k-1}]_{2}}$-code

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, 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 ${\displaystyle k}$ to codewords of length ${\displaystyle 2^{k}}$. It is unique in that each non-zero codeword has a Hamming weight of exactly ${\displaystyle 2^{k-1}}$, which implies that the distance of the code is also ${\displaystyle 2^{k-1}}$. In standard coding theory notation for block codes, the Hadamard code is a ${\displaystyle [2^{k},k,2^{k-1}]_{2}}$-code, that is, it is a linear code over a binary alphabet, has block length ${\displaystyle 2^{k}}$, message length (or dimension) ${\displaystyle k}$, and minimum distance ${\displaystyle 2^{k}/2}$. 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 ${\displaystyle [2^{k},k+1,2^{k-1}]_{2}}$-code and thus has a slightly better rate while maintaining the relative distance of ${\displaystyle 1/2}$, 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.

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