Consider the binary linear code which adds three parity bits $3,4,5$ to two information

bits $1,2$ as follows: $3=1,4=1+2,5=2.$

a$)(1$ marks$)$ List all its codewords.

b$)(1$ mark$)$ Determine its minimum distance and state its error correction capability.

c$)(2$ marks$)$ Determine its generator matrix and parity check matrix.

d$)(1$ mark$)$ Suppose the vector $(1,0,1,0,1)$ is received. Compute the syndrome.

e$)(2$ marks$)$ Use the syndrome to identify the error bit, if any. Explain your reasoning.