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 ?
For the given linear code (C) over (GF(2)) with generator matrix [ G=\begin{bmatrix} 1&0&1&1&1&0\ 0&1&1&0&1&0\ 0&0&1&1&0&1 \end{bmatrix}, ] the dimension is (k=3) and the length is (n=6). To obtain a parity check matrix, first convert (G) to systematic form using elementary row operations over (GF(2)). Add the third row to ___ __________ _______ ______ _____ ________.
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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.
For each of the linear codes, find the degree, a generator matrix and a parity check matrix.
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.
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 |
Factor x5 - 1 over . Give the generator polynomials of all cyclic codes of length five over
.
Which of the following statements are true and which are false? Justify your answer with a short proof or a counterexample.
i) If the weight of each element in the generating matrix of a linear code is at least r, the minimum distance of the code is at least r.
ii) There is no linear self orthogonal code of odd length.
iii) There is no 3-cyclotomic coset modulo 121 of size 25.
iv) There is no duadic code of length 15 over .
v) There is no LDPC code with parameters ,
and
.
Find the image of the circle under the mapping
. What happens when
?