Introduction to Cryptography with Coding Theory, 3rd Edition

24.13 Computer Problems

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

   $$\begin{array}{llllllllllllllllllllllll}(0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1)\text{,}\hfill \\ (0\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 0)\text{,}\hfill \\ (1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 0\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1\text{,}\hfill & 1)\text{.}\hfill \end{array}$$

   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

   $(0\text{,}1\text{,}1\text{,}0\text{,}0\text{,}0\text{,}1\text{,}1\text{,}0\text{,}0\text{,}0\text{,}1\text{,}0\text{,}1\text{,}0)$

   is received. Assuming there is at most one error, correct it and determine the original 11-bit message. ...