 The Hadamard code is a linear code, and all linear codes can be generated by a generator matrix formula_32.This is a matrix such that formula_33 holds for all formula_14, where the message formula_21 is viewed as a row vector and the vector-matrix product is understood in the vector space over the finite field formula_36.In particular, an equivalent way to write the inner product definition for the Hadamard code arises by using the generator matrix whose columns consist of "all" strings formula_24 of length formula_8, that is,

where formula_40 is the formula_41-th binary vector in lexicographical order.