Problem sheet 6 2005, Feb. 17th

MT361 Error correcting codes

Ex. 1 There exists a perfect [23, 12, 7] code. Determine the number of codewords of weight 0, 1, 2, 3, …, 23 (use the code is perfect!) From this determine the probability p_{correct} of correct decoding a word of length 23.

Ex. 2 Prove: Suppose C is a linear [n, k]-code having generator matrix G. Then a vector v⃗ ∈ V (n, q) belongs to C^{⊥} ⇔ v⃗G^{⊤} = 0, where G^{⊤} denotes the transpose of G.

Ex. 3 Determine the dual codes C_{1}^{⊥} and C_{2}^{⊥} for C_{1} = {000, 111} and C_{2} = {00000, 11111}.

Ex. 4 Prove that if E_{n} is the binary even weight code of length n, then E_{n}^{⊥} is the repetition code of lengt n. Give the generator matrices and the parity check matrices for the two codes.

Ex. 5 Let C be the ternary linear code with generator matrix

[ \begin{matrix}1 & 1 & 1 & 0 \\ 2 & 0 & 1 & 1 \end{matrix} ] .

a) Find a generator matrix G for C in standard form

b) Find a parity check matrix H for C in standard form.