The [15,11] Hamming code has parity matrix H =
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The columns of H are all the non-zero binary vectors of length 4, which means that each column is a unique binary representation of an integer between 1 and 15. Show more…
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The Hamming (4,7)-code is defined by means of the parity check matrix H and/or the generator matrix G, where H = [0 0 0 1 1 1 1; 0 1 1 0 0 1 1; 1 0 1 0 1 0 1], G = [1 0 0 0 0 1 1; 0 1 0 0 1 0 1; 0 0 1 0 1 1 0; 0 0 0 1 1 1 1]. (a) Encode the vector u = (1 1 0 0); (b) Decode the vectors v = (1 0 1 1 0 1 0) and w = (0 0 0 0 1 1 1). (c) Find all strings of length 7 which are decoded to (1 1 0 0).
Sri K.
Example 1.3.2: One generator matrix for the [7, 4] Hamming code H is presented in Example 1.2.3. Let Tz be the code of length 8 and dimension obtained from H by adding an overall parity check coordinate t0 to each vector of G and thus t0 to each codeword of Hz. Then G is a generator matrix for Ts. It is easy to verify that Ts is a self-dual code. Example 1.3.3: The [4, 2] ternary code Hs,2, often called the tetracode, has a generator matrix G in standard form, given by G = [' %1 -4]. Ts code is also self-dual.
Suman K.
The 4 by 4 Hadamard matrix is entirely $+1$ and $-1$ : $$ H=\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ 1 & -1 & -1 & 1 \end{array}\right] $$ Find $H^{-1}$ and write $v=(7,5,3,1)$ as a combination of the columns of $H$.
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