Systematic code

A systematic code is a code which assigns a code word of length n to each message word of length k , the message word being explicitly part of the assigned code word.

As an example, apply parity checks in which the message word one or more check bits are appended.

example

A systematic [n, k] code can look like this: . This code could be achieved by following function describe: . And in detail we would have: ${\displaystyle C=\{(000),(011),(101),(110)\}}$ ${\displaystyle f:\{0,1\}^{2}\to \{0,1\}^{3}}$

${\displaystyle \,(00)\mapsto (000)}$, , , . ${\displaystyle \,(01)\mapsto (011)}$ ${\displaystyle \,(10)\mapsto (101)}$ ${\displaystyle \,(11)\mapsto (110)}$

Now we have a systematic in k point code of length n with minimum distance d ( Hamming distance ) and can also write: . ${\displaystyle [3,2,2]-Code}$