Systematic Code is a #Block Code in which message bits are transmitted in unaltered form. The use of Systematic Code simplifies the implementation of #decoder.
For example, if the resulting Generator Matrix or Parity-Check Matrix is not systematic, we could XOR the non-systematic part out with rows that exhibits systematic characteristic.
$$ G' = \left[ \begin{array}{ccccccc} 1 & 1 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{array} \right] $$
Row 3 and 4 is not systematic as it doesn’t fulfil the property of being a Generator Matrix where \(G = [P|I^k]\) as there are discrepancies in Identity Matrix. Here, we XOR Row 3 with Row 1 in order to neutralise the 5th bit (which is 1) in order to make it systematic. Following the steps, we could get the final result systematic \(G\):
$$ G = \left[ \begin{array}{ccccccc} 1 & 1 & 0 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 1 & 0 & 0\\ 1 & 1 & 1 & 0 & 0 & 1 & 0\\ 1 & 0 & 1 & 0 & 0 & 0 & 1 \end{array} \right] $$