 In communication theory, this is simply called the Hadamard code and it 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.