91Ó°ÊÓ

For each of the following encoding functions, find the minimum distance between the code words. Discuss the errordetecting and error-correcting capabilities of each code. a) \(\begin{array}{rl}E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{5} \\\ 00 \rightarrow 00001 & 01 \rightarrow 01010 \\ 10 \rightarrow 10100 & 11 \rightarrow 11111\end{array}\) b) \(E: \mathbf{Z}_{2}^{2} \rightarrow \mathbf{Z}_{2}^{10}\) \(\begin{array}{cl}00 \rightarrow 0000000000 & 01 \rightarrow 0000011111 \\ 10 \rightarrow 1111100000 & 11 \rightarrow 1111111111\end{array}\) \begin{array}{ll} \text { c) } E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{6} & \\ 000 \rightarrow 000111 & 001 \rightarrow 001001 \\ 010 \rightarrow 010010 & 011 \rightarrow 011100 \\ 100 \rightarrow 100100 & 101 \rightarrow 101010 \\ 110 \rightarrow 110001 & 111 \rightarrow 111000 \\ \text { d) } E: \mathbf{Z}_{2}^{3} \rightarrow \mathbf{Z}_{2}^{8} & \\ 000 \rightarrow 00011111 & 001 \rightarrow 00111010 \\ 010 \rightarrow 01010101 & 011 \rightarrow 01110000 \\ 100 \rightarrow 10001101 & 101 \rightarrow 10101000 \\ 110 \rightarrow 11000100 & 111 \rightarrow 11100011 \end{array}

Let \(G\) be a group with subgroups \(H\) and \(K\). If \(|G|=660\), \(|K|=66\), and \(K \subset H \subset G\), what are the possible values for | \(H \mid ?\)

a) In how many ways can the seven (identical) horses on a carousel be painted with black, brown, and white paint in such a way that there are three black, two brown, and two white horses? b) In how many ways would there be equal numbers of black and brown horses? c) Give a combinatorial argument to verify that for all \(n \in \mathbf{Z}^{+}, n^{7}+6 n\) is divisible by 7

a) In how many ways can we paint the eight squares of a \(2 \times 4\) chessboard, using the colors red and white? (The back of the chessboard is black cardboard.) b) Find the pattern inventory for the colorings in part (a). c) How many of the colorings in part (a) have four red and four white squares? How many have six red and two white squares?

In \(S_{5}\) find an element of order \(n\), for all \(2 \leq n \leq 5\). Also determine the (cyclic) subgroup of \(S_{5}\) that each of these elements generates.

Let \(H\) and \(K\) be subgroups of a group \(G\), where \(e\) is the identity of \(G\). a) Prove that if \(|H|=10\) and \(|K|=21\), then \(H \cap K=\\{e\\}\). b) If \(|H|=m\) and \(|K|=n\), with \(\operatorname{gcd}(m, n)=1\), prove that \(H \cap K=\\{e\\}\)