Information theory and coding

Homework $2$

Answer the following questions submit the answer online using moodle:

Do not use Al $,$ do not use online resources. this will result in a zero mark

Q$1$:

For a $(6,3)$ systematic linear block code, the codeword comprises m$0$m$1$m$2$p$0$p$1$p$2$ where the $3$ parity

$c_0=m_{0}bigoplus{m}_1,c_1=m_{0}bigoplus{m}_2,c_2=m_{1}bigoplus{m}_2$

check bits p$0$p$1$p$2$ are formed from the information bits as follows:

Find

i$.$ The parity check matrix

ii$.$ The generator matrix

iii. All possible codewords

iv$.$ The minimum weight

v$.$ Minimum distance

vi$.$ Error detecting & correcting capability of this code

vii. If the received sequence is $101000,$ calculate the syndrome and decode the received sequence

Q$2$: If G and H are generator and parity$-$check matrices, show that $G{H}^T=0.$