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===Construction using inner products===
When given a binary message <math>x\in\{0,1\}^k</math> of length <math>k</math>, the Hadamard code encodes the message into a codeword <math>\text{Had}(x)</math> using an encoding function <math>\text{Had} : \{0,1\}^k\to\{0,1\}^{2^k}.</math>
This function makes use of the [[inner product]] <math>\langle x , y \rangle</math> of two vectors <math>x,y\in\{0,1\}^k</math>, which is defined as follows:
:<math>\langle x , y \rangle = \sum_{i=1}^{k} x_i y_i\ \bmod\ 2\,.</math>
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