24.13 Computer Problems – Introduction to Cryptography with Coding Theory, 3rd Edition

24.13 Computer Problems

  1. Three codewords from the Golay code G24 are sent and you receive the vectors

    (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).

    Correct the errors. (The Golay matrix is stored as golay and the matrix B is stored in the downloadable computer files (bit.ly/2JbcS6p) as golaybt.)

  2. 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)

    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.)