Question
Let and
be two binary codes with generator matrices
respectively.
i) Find the minimum distance of both the codes.
Table 1: Table for F16.
| 0000 | 0 | 1000 | α³ | 1011 | α⁷ | 1110 | α¹¹ |
|---|---|---|---|---|---|---|---|
| 0001 | 1 | 0011 | α⁴ | 0101 | α⁸ | 1111 | α¹² |
| 0010 | α | 0110 | α⁵ | 1010 | α⁹ | 1101 | α¹³ |
| 0100 | α² | 1100 | α⁶ | 0111 | α¹⁰ | 1001 | α¹⁴ |
ii) Find the generator matrix of the code
obtained from C1 and C2 by (u|u+v) construction. Also, find the minimum distance of C .
Answer :
Word Count : 667
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For the given problem, we solve step by step. We are given: [ G_1 = \begin{bmatrix} 1 & 0 & 0 & 1\ 0 & 1 & 0 & 0\ 0 & 0 & 1 & 1 \end{bmatrix}, \quad G_2 = \begin{bmatrix} 1 & 0 & 0 & 1\ 0 & 1 & 1 & 0 \end{bmatrix} ] --- i) Minimum distance of (\mathcal{C}_1) and (\mathcal{C}_2): * The minimum distance (d_{\min}) of a linear code is the smallest weight of a nonzero codeword. Step 1: Generate all codewords of (\mathcal{C}_1) (\mathcal{C}_1) has dimension (k_1 = 3), length (n_1 = 4). Let (\mathbf{u} = (u_1,u_2,u_3)). Then codewords are: [ \mathbf{c}_1 = \mathbf{u} G_1 ] Compute all 8 possible codewords: 1. ( (0,0,0) G_1 = 0000 ) → weight 0 2. ( (1,0,0) G_1 = 1\ 0\ 0\ 1 = 1001 ) → weight 2 3. ( (0,1,0) G_1 = 0\ 1\ 0\ 0 = 0100 ) → weight 1 4. ( (0,0,1) G_1 = 0\ 0\ 1\ _____ _____ ___ ____ _______ __________ _____ ________ _______ ______ ___.
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