91Ó°ÊÓ

Let \(G=S_{4}\). (a) For \(\alpha=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 3 & 4 & 1\end{array}\right)\), find the subgroup \(H=\langle\alpha\rangle\). (b) Determine the left cosets of \(H\) in \(G\).

A binary symmetric channel has probability \(p=0.05\) of incorrect transmission. If the code word \(c=011011101\) is transmitted, what is the probability that (a) we receive \(r=\) 011111101 ? (b) we receive \(r=111011100\) ? (c) a single error occurs? (d) a double error occurs? (e) a triple error occurs? (f) three errors occur, no two of them consecutive?

Express each of the following elements of \(S_{7}\) as a product of disjoint cycles. $$ \begin{aligned} \alpha &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 4 & 6 & 7 & 1 & 5 & 3 \end{array}\right) \\ \beta &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 6 & 5 & 2 & 1 & 7 & 4 \end{array}\right) \\ \gamma &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 2 & 3 & 1 & 7 & 5 & 4 & 6 \end{array}\right) \\ \delta &=\left(\begin{array}{lllllll} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 4 & 2 & 7 & 1 & 3 & 6 & 5 \end{array}\right) \end{aligned} $$

Why is the set \(\mathbf{Z}\) not a group under subtraction?

Let \(E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{9}\) be the encoding function for the \((9,3)\) triple repetition code. a) If \(D: \mathbf{Z}_{2}^{9} \rightarrow \mathbf{Z}_{2}^{3}\) is the corresponding decoding function, apply \(D\) to decode the received words (i) 111101100 ; (ii) 000100011 ; (iii) 010011111 . b) Find three different received words \(r\) for which \(D(r)=\) \(000 .\) c) For each \(w \in \mathbf{Z}_{2}^{3}\), what is \(\left|D^{-1}(w)\right| ?\)

a) Construct a decoding table (with syndromes) for the group code given by the generator matrix $$ G=\left[\begin{array}{lllll} 1 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \end{array}\right] $$ b) Use the table from part (a) to decode the following received words. \(\begin{array}{cccc}11110 & 11101 & 11011 & 10100 \\ 10011 & 10101 & 11111 & 01100\end{array}\) c) Does this code correct single errors in transmission?

If \(G\) is a group of even order, prove that there is an element \(a \in G\) with \(a \neq e\) and \(a=a^{-1}\).

Let $$ H=\left[\begin{array}{lllllll} 1 & 1 & 0 & 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 0 & 0 & 1 \end{array}\right] $$ be the parity-check matrix for a Hamming \((7,4)\) code. a) Encode the following messages: \(\begin{array}{llllll}1000 & 1100 & 1011 & 1110 & 1001 & 1111 .\end{array}\) b) Decode the following received words: \(\begin{array}{cccc}1100001 & 1110111 & 0010001 & 0011100\end{array}\) c) Construct a decoding table consisting of the syndromes and coset leaders for this code. d) Use the result in part (c) to decode the received words given in part (b).

The \((5 m, m)\) five-times repetition code has encoding function \(E: \mathbf{Z}_{2}^{m} \rightarrow \mathbf{Z}_{2}^{5 m}\), where \(E(w)=w w w w w .\) Decoding with \(D: \mathbf{Z}_{2}^{5 m} \rightarrow \mathbf{Z}_{2}^{m}\) is accomplished by the majority rule. (Here we are able to correct single and double errors made in transmission.) a) With \(p=0.05\), what is the probability for the transmission and correct decoding of the signal \(0 ?\) b) Answer part (a) for the message 110 in place of the signal \(0 .\) c) For \(m=2\), decode the received word $$ r=0111001001 $$ d) If \(m=2\), find three received words \(r\) where \(D(r)=00\). e) For \(m=2\) and \(D: \mathbf{Z}_{2}^{10} \rightarrow \mathbf{Z}_{2}^{2}\), what is \(\left|D^{-1}(w)\right|\) for each \(w \in \mathbf{Z}_{2}^{2} ?\)

Let \(G=\\{q \in \mathbf{Q} \mid q \neq-1\\}\). Define the binary operation \(\circ\) on G by \(x \circ y=x+y+x y\). Prove that \((G, \circ)\) is an abelian group