Polynomial code

[fredo07]

Hello!

So i'm having trouble understanding how you would use a generator polynomial to generate binary digits of codewords.
The polynomial generator code is given as G(x)=x^4+x^2+x+1
From what i understand you need the degree which is 4 and you need to choose an integer n>m.
I've only read an example on wikipedia and i'm having trouble understanding the more complex explanations :-?

I'd appreciate any help and thank you very much for taking the time to read my post!
this is the only link i've read that i "kind-off" understand!?
http://en.wikipedia.org/wiki/Polynomial_code

 

[Ishuda]

The example I refer to below is from the link you give.

First, since you are generating binary digits, i.e. either 0 or 1, the field is, as given in the example, GF(2).

Next, as you mentioned, choose an integer n \(\displaystyle \ge\) m. In the example the degree of the generating polynomial g(x) is m=4 for you and m=2 for the example. The degree of the code word polynomial is n is undetermined for you and n = 5 for the example.

Next, what is the highest degree of a polynomial, = d₁, you can multiply g(x) by and have the resulting degree, =d₂, remain less than n; d₂ is undetermined but equal to n-1 for you and 4 in the example. Since d₁ is d₂ minus m, yours would be n - 5 and is 2 in the example.

Next, how many polynomials are there of degree less than of equal to d₁ if you can only choose coefficients from the given field. Well you can choose q (the number of elements of the field) for the zero^th power, q for first power, q for the second power, ...., and q for d₁^th power, that is a total of (d₁ + 1) q's or q^(d₁+1) or 2^n-4 for you and 2³ = 8 for the example.

Finally, what are those polynomials and what are the resulting polynomials (code words)? The example lists them. Looking at one of the resulting polynomials in the example, say x² + x + 1, we write that as
x² + x + 1 = 0 x⁴ + 0 x³ + 1 x² + 1 x¹ + 1 x⁰
The code word is the coefficients in order or 00111.

Edit: Note that when d₁ gets large, it is much better to have a computer to figure out what the polynomials and resulting polynomials are than trying to do it yourself. A computer could use n = 100 and m = 85 resulting in about a '32768 word vocabulary' in a few seconds which could form a 'one time pad'. Since, even if you restricted yourself to a generating polynomial of degree 50 up to 85, you could change your vocabulary every second and still have enough to last over a thousand years.