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

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.

• That depends exclusively on your definition of "systematic". We don't know that. Different definitions exist. Dec 9, 2022 at 21:20
• (the differences are exactly in whether only 1, only 2, or all 1&2&3 describe systematic codes. The information theorists tend to say that all three are the same code anyways, just different encoders. So, you need to read or infer the definition that exactly the material you're currently working with uses. Usually, it makes no difference at. Sometimes the ordering does. Gotta use the definitions.) Dec 9, 2022 at 21:27
• @MarcusMüller 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 has given below the statement in which parity bits are left of the message bits. Dec 10, 2022 at 6:46
• I'm altering your question to actually contain the source of your confusion, then! Dec 10, 2022 at 10:11

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.

• Thanks. I just read in MIT-page-12 that any linear code can be transformed into an equivalent systematic code. Does that mean, suppose a code alter the message bits, there is an equivalent systematic form that doesn't alter the message bits? Dec 11, 2022 at 5:49
• You can read English just as well as me. Dec 11, 2022 at 7:54
• Seriously, you're just asking me to repeat the same procedure as I demonstrated above. You have a statement. You wonder whether it applies to something. So: Try applying it. If it applies, the statement will be correct. I won't do that for you. Dec 11, 2022 at 11:41
• Mmm. English is not my 1st language. Sometimes I loose important information embedded in a single word of the statement. Many YouTube videos were explaining systematic code as one with message bits in the beginning or at the end. They skipped other possibilities. Now I have clarity. Thanks. Dec 11, 2022 at 13:37
• general rule of thumb: stay away from youtube. Dec 11, 2022 at 14:01