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24.13 Computer Problems

Three codewords from the Golay code $G_{24}$ $G_{24}$ are sent and you receive the vectors

$\begin{array}{l} (0, & 1, & 0, & 0, & 0, & 0, & 1, & 1, & 0, & 1, & 0, & 1, & 1, & 0, & 1, & 1, & 0, & 1, & 0, & 0, & 0, & 0, & 1, & 1), \\ (0, & 0, & 1, & 0, & 1, & 0, & 1, & 0, & 1, & 0, & 1, & 0, & 0, & 1, & 0, & 0, & 1, & 1, & 1, & 0, & 1, & 1, & 0, & 0), \\ (1, & 1, & 1, & 0, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 0, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1) . \end{array}$ $\begin{array}{l} (0, & 1, & 0, & 0, & 0, & 0, & 1, & 1, & 0, & 1, & 0, & 1, & 1, & 0, & 1, & 1, & 0, & 1, & 0, & 0, & 0, & 0, & 1, & 1), \\ (0, & 0, & 1, & 0, & 1, & 0, & 1, & 0, & 1, & 0, & 1, & 0, & 0, & 1, & 0, & 0, & 1, & 1, & 1, & 0, & 1, & 1, & 0, & 0), \\ (1, & 1, & 1, & 0, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 0, & 1, & 1, & 1, & 1, & 1, & 1, & 1, & 1) . \end{array}$

Correct the errors. (The Golay matrix is stored as golay and the matrix $B$ $B$ is stored in the downloadable computer files (bit.ly/2JbcS6p) as golaybt.)
An 11-bit message is multiplied by the generating matrix for the Hamming [15, 11] code and the resulting codeword is sent. The vector

$(0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0)$ $(0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0)$

is received. Assuming there is at most one error, correct it and determine the original 11-bit message. (The parity check matrix for the Hamming [15, 11] code is stored in the downloadable computer files (bit.ly/2JbcS6p) as hammingpc.)

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Table of Contents for 24.13 Computer Problems

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Table of Contents for
24.13 Computer Problems