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Normally, Hadamard codes are based on [[Hadamard matrix#Sylvester's construction|Sylvester's construction of Hadamard matrices]], but the term “Hadamard code” is also used to refer to codes constructed from arbitrary [[Hadamard matrix|Hadamard matrices]], which are not necessarily of Sylvester type.
In general, such a code is not linear.
Such codes were first constructed by [[
If ''n'' is the size of the Hadamard matrix, the code has parameters <math>(n,2n,n/2)_2</math>, 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.
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