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 .
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For a linear code C with generator matrix
find the parity check matrix. Check that any two columns of the parity check matrix are linearly independent and there are three columns that are linearly dependent. What is the minimum distance of C ?
Let C be the [5, 2] binary code generated by
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Find the zeros and singularities of the function in
. Also find the residue at the poles
Find all the singularities of the function .
) Let C be the ternary [8,3] narrow-sense BCH code of designed distance , which has defining set
. Use the primitive root 8th root of unity you chose in 4a) to avoid recomputing the the table of powers. If
is the generator polynomial of C and
is the received word, find the transmitted codeword. Use the following table in 1.
Evaluate where c is the circle
.
Which of the following binary codes are linear?
i)
ii)
Justify your answer.