# Reed Solomon encoding

Started by June 10, 2006
```Here I post the same message I've posted in another forum, where I've

I've a basic question about encoding with Reed Solomon code.  Suppose
that C is a RS-code of length n = q-1 over the field F_q, designed
distance d and dimension k. Let b a primitive element in F_q. The
generator of the code is g(x) = (x-b) (x-b^2)...(x-b^(d-1)). Now, if I
want to encode an element p(x) in F_q [x] of degree less or equal to
k-1, I must multiply g(x) p(x). The result is an element of C. Is it
correct to encode in this way? (I suppose the answer is affermative).
different. And the result of encoding too. So, if a choose a fixed
RS-code C, with a generator g(x), using the first method and the
systematic encoding I have two different result. Can you confirm this?

```
```sugolini@gmail.com said the following on 10/06/2006 22:51:
> I've a basic question about encoding with Reed Solomon code.  Suppose
> that C is a RS-code of length n = q-1 over the field F_q, designed
> distance d and dimension k. Let b a primitive element in F_q. The
> generator of the code is g(x) = (x-b) (x-b^2)...(x-b^(d-1)). Now, if I
> want to encode an element p(x) in F_q [x] of degree less or equal to
> k-1, I must multiply g(x) p(x). The result is an element of C. Is it
> correct to encode in this way? (I suppose the answer is affermative).
> However I have read about a systematic encoding. This method seems
> different. And the result of encoding too. So, if a choose a fixed
> RS-code C, with a generator g(x), using the first method and the
> systematic encoding I have two different result. Can you confirm this?

The systematic form of any coding scheme (not just RS) maintains the
same codeword set, but does not maintain the mapping between the set of
possible inputs and this codeword set.

So yes, you will get a different output if you adopt a systematic
encoder.  However, as the codeword set is the same, your Hamming and
Euclidean distance spectra will be the same as for the non-systematic
form, so there is no change to the effectiveness of the code.

--
Oli
```
```Oli Filth wrote:

> The systematic form of any coding scheme (not just RS) maintains the
> same codeword set, but does not maintain the mapping between the set of
> possible inputs and this codeword set.
>
> So yes, you will get a different output if you adopt a systematic
> encoder.  However, as the codeword set is the same, your Hamming and
> Euclidean distance spectra will be the same as for the non-systematic
> form, so there is no change to the effectiveness of the code.
>
>
>
> --
> Oli

Thank you a lot.

```