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Hamming codes are designed to correct a single error. A valid codeword $p_1 p_2 m_3 p_4 m_5 m_6 m_7$ satisfies the constraints \begin{align} &p_1 \oplus m_3 \oplus m_5 \oplus m_7 = 0 \\ &p_2 \oplus m_3 \oplus m_6 \oplus m_7 = 0 \\ &p_4 \oplus m_5 \oplus m_6 \oplus m_7 = 0 \end{align} (This is arranged simply by choosing $p_1 = m_3 \oplus m_5 \...


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