What are the possible forms of generator matrix of a systematic linear block code?

Question

My text book, Communication systems by Simon Haykin says

Block codes in which the message bits are transmitted in unaltered form are called systematic codes.

I am not getting what it means. A diagram is given below the statement in which parity bits are left of the message bits.

Are the following forms valid?

1)\begin{bmatrix} P |I \\ \end{bmatrix}

2)\begin{bmatrix} I|P \\ \end{bmatrix}

3)\begin{bmatrix} p_{1} & p_{2} & p_{3} & I & p_{4} & p_{5} \\ \end{bmatrix}

$p_1$ , $p_2$, $p_3$ are column vectors of parity generator matrix, $I$ is the identity matrix.

Answer 1

Block codes in which the message bits are transmitted in unaltered form are called systematic codes.

This is relatively explicit: If the message bits are

1. transmitted themselves
2. and that in unaltered form

is called systematic in your textbook.

Let's check that for your three examples:

1. $[P|I]$: Systematic, because
   1. The message bits are indeed transmitted,
   2. the entries in the identity do not alter the bits.
2. $[I|P]$: Systematic, because
   1. The message bits are indeed transmitted,
   2. the entries in the identity do not alter the bits.
3. $[p_{i_1} p_{i_2} \ldots p_{i_j} I p_{i_{j+1}} \ldots p_{i_{n-k}}]$: Systematic, because
   1. The message bits are indeed transmitted,
   2. the entries in the identity do not alter the bits.

Just apply the words of the definition down to the letter. You don't even have to have one contiguous $I$ in there according to the definition. The message bits just need to be all transmitted in unaltered form – at any place, in any order.

From a practical perspective, your three options really really make no difference at all to any system: they're just permutation of code word bit positions. Think about that – that's just exactly what an interleaver does, anyway. And since permutation is an invertible operation, each code bit, each symbol of the code word has the same properties (e.g. error probability) as before and carries exactly the same amount of Shannon information – it's really the same code. Just written down in a different order.