Consider the binary linear code which adds three parity bits c$1,$ c$2,$ c$3,$ c$4,$ c$5$ to two information

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

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.