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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Evaluate the following integrals
a) .
b) .
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 |
Evaluate where c is the circle
.
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.
Let C be a cyclic code of length eight over with generator polynomial
. Find the generator matrix of C, the generator polynomial of C and the parity check matrix of C.
Find the radius of convergence of the following series.
i)
ii)
Find the maximum modulus of on the closed circular region defined by
.