Let C be a systematic linear block code with the following parity−$-$check matrix,

H=[[10001011$=[[10001011$; 01001110$01001110$; 00100111$00100111$; 00011111]]$00011111]]$

($($a)$)$ Find the code rate of C.(3$.(3$ marks)($)($b)$)$ Find the generator matrix of C.(3$.(3$ marks)($)($c)$)$ Find the codeword in C for message bits ([0110]).(3$([0110]).(3$ marks)($)($d)$)$ Find the minimum distance of this code.

(3$(3$ marks)$)$

(3$(3$ marks)$)$

this received vector. Using syndrome decoding, find the position of this error in r.(5$.(5$ marks)($)($g)$)$ Design the parity−$-$check matrix for a new (7,4)$(7,4)$ systematic linear block code by eliminating one row and one column in H,$,$ such that the new code has the largest minimum distance.