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 .
The code (\mathcal C_1) has generator matrix [ G_1=\begin{bmatrix} 1&0&0&1\ 0&1&0&0\ 0&0&1&1 \end{bmatrix}. ] All codewords are obtained as linear combinations of the rows. The nonzero codewords are [ (1,0,0,1),\ (0,1,0,0),\ (0,0,1,1), ] their pairwise sums [ (1,1,0,1),\ (1,0,1,0),\ (0,1,1,1), ] and the sum of all three [ (1,1,1,0). ] Their Hamming weights ___ __________ _________ _____ __________.
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he aim of this exercise is to show that every binary repetition code of odd length is perfect.
i) Find the value of t and d for a binary repetition code of length . \hfill (2)
ii) Show that
(Hint: Start with the relation
iii) Deduce that every repetition code of odd length is perfect.
Show that there is only one linear fractional transformation that maps three given distinct points z1, z2 and z3 in the extended z plane onto three specified distinct points w1, w2 and w3 in the extended w plane.
Let C be the [5, 2] binary code generated by
Find the weight enumerator C of C . Use McWilliams identity to find the weight enumerator of C. Verify your answer by finding the generator matrix of C and finding the weight distribution of C.
Evaluate , where c is the eight like figure shown in Fig. 1.
Evaluate where c is the circle
.
Find the minimum distance for each of the codes.
Find all the singularities of the function .