Hadamard code: Difference between revisions

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The Hadamard code is an example of a [[linear code]] over a [[binary set|binary alphabet]] that maps messages of length <math>k</math> to codewords of length <math>2^k</math>.

It is unique in that each non-zero codeword has a [[Hamming weight]] of exactly <math>2^{k/2-1}</math>, which implies that the [[Block code#The distance d|distance]] of the code is also <math>2^{k/2-1}</math>.

In standard [[Block code#Popular notation|coding theory notation]] for [[block code]]s, the Hadamard code is a <math>[2^k,k,2^{k/2-1}]_2</math>-code, that is, it is a [[linear code]] over a [[binary set|binary alphabet]], has [[Block code#The block length n|block length]] <math>2^k</math>, [[Block code#The message length k|message length]] (or dimension) <math>k</math>, and [[Block code#The distance d|minimum distance]] <math>2^k/2</math>.

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 <math>[2^{k-1},k,2^{k-2}]_2</math>-code and thus has a slightly better [[Block code#The rate R|rate]] while maintaining the relative distance of <math>1/2</math>, and is thus preferred in practical applications.