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Chapter 16: Problem 6
Compare the rates of the Hamming \((7,4)\) code and the \((3,1)\) triple- repetition code.
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Given \(n \in \mathbf{Z}^{+}\), let the set \(M(n, k) \subseteq \mathbf{Z}_{2}^{n}\) contain the maximum number of code words of length \(n\), where the minimum distance between code words is \(2 k+1\). Prove that $$ \frac{2^{n}}{\sum_{l=0}^{2 k}\left(\begin{array}{l} n \\ l \end{array}\right)} \leq|M(n, k)| \leq \frac{2^{n}}{\sum_{l=0}^{k}\left(\begin{array}{l} n \\ \imath \end{array}\right)} $$ (The upper bound on \(|M(n, k)|\) is called the Hamming bound; the lower bound is referred to as the Gilbert bound.)
Determine the plaintext for the RSA ciphertext 09863029 \(\begin{array}{lllllllll}1134 & 1105 & 1232 & 2281 & 2967 & 0272 & 1818 & 2398 & 1153, \text { if }\end{array}\) \(e=17\) and \(n=3053\).
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).
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 \(H \cap K=\\{e\\}\)
a) How many rigid motions (in two or three dimensions) are there for a regular pentagon? Describe them geometrically. b) Answer part (a) for a regular \(n\)-gon, \(n \geq 3\).
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