This is because, if the first bit of formula_24 is zero, formula_25, then the inner product contains no information whatsoever about formula_26, and hence, it is impossible to fully decode formula_21 from those positions of the codeword alone. On the other hand, when the codeword is restricted to the positions where formula_28, it is still possible to fully decode formula_21. Hence it makes sense to restrict the Hadamard code to these positions, which gives rise to the "augmented" Hadamard encoding of formula_21; that is, formula_31. 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. For example, the generator matrix for the Hadamard code of dimension formula_42 is:  The matrix formula_32 is a formula_45-matrix and gives rise to the linear operator formula_46. The generator matrix of the "augmented" Hadamard code is obtained by restricting the matrix formula_32 to the columns whose first entry is one. For example, the generator matrix for the augmented Hadamard code of dimension formula_42 is:  Then formula_50 is a linear mapping with formula_51. For general formula_8, the generator matrix of the augmented Hadamard code is a parity-check matrix for the extended Hamming code of length formula_4 and dimension formula_54, which makes the augmented Hadamard code the dual code of the extended Hamming code. Hence an alternative way to define the Hadamard code is in terms of its parity-check matrix: the parity-check matrix of the Hadamard code is equal to the generator matrix of the Hamming code. Hadamard codes are obtained from an "n"-by-"n" Hadamard matrix "H". In particular, the 2"n" codewords of the code are the rows of "H" and the rows of −"H".