1 Problem sheet 4 2006

MT3610/5461 Error correcting codes

Ex. 1 Given a binary (11, 24, 5)-code. Argue why this code cannot be linear.

Ex. 2 Let E_{n} be the set of all elements of V (n, 2) with even weight. Show that E_{n} is a linear code. What are its parameters [n,k,d]? What is M? Write down a generator matrix in standard form.

Ex. 3 Prove that, in a binary linear code, either all the codewords have even weight or exactly half have even weight and half have odd weight.

Ex. 4 Let C_{1} and C_{2} be binary linear codes having the generator matrices

G_{1} = [ \begin{matrix}1 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 \end{matrix} ] and G_{2} = [ \begin{matrix}1 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 & 1 \end{matrix} ]. List the code-

words of C_{1} and C_{2} and hence find the minimum distance of each code.