 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.