Matrix of Galois Field Elements

Hi There,

I need to solve Ax=b in an Galois Field environment and to my frustration I cannot get Matlab to do this simple task. My current problem is setting up the A matrix. Each element in the A Matrix = the exponentiation of an Galois field element( gf(GenRoots)) with a power, dictated by PolyPowers(Polynomial Powers).

K= 1 ;
n=255 ;
k=127;
for GenRoots = (1+K):(K+n-k)
A_Row = GenRoots - K ;   
    for PolyPowers = 1:128
        A_Col = PolyPowers;
        A(A_Row,A_Col) = (gf(GenRoots,4).^(PolyPowers+127))
    end
end

I believe I'm buggering the syntax and just not suing the commands correctly but I cannot figure out what I am doing wrong.

The Matlab errors I get are:

??? The following error occurred converting from gf to double:
Error using ==> double
Conversion to double from gf is not possible.

Error in ==> test at 16
        A(A_Row,A_Col) = (gf(GenRoots,4).^(PolyPowers+127))

How can I get an invertible matrix A, filled with galois field elements, after I have done some arithmetic with them?

Any help and comments will greatly be appreciated.

Thanks for your time,

Rudolf

Answer 1

[Andrew Newell]

You must have a different version of MATLAB than I do (2010b). The error message I get is:

??? Error using ==> gf.gf at 66
X must be between 0 and 2^m-1

I get the same error message if I try this:

GenRoots = 16;
gf(GenRoots,4)

The solution is to use mod:

gf(mod(GenRoots,2^4),4)

EDIT: Similarly, in your loop you need

gf(mod(GenRoots,2^4),4).^(PolyPowers+127)

Note that I am creating the gf element first and then taking the power - with such large powers, you'll get an Inf otherwise!

EDIT 2: On my machine, this code does the same thing as your loop but 2000 times faster:

A1 = gf(repmat(mod((2:129)',2^4),1,128),4);
pp = repmat(1:128,128,1)+127;
A1 = A1.^pp;

[Rudolf Hoehler]

Thank you very much for the detailed reply! This helps a lot!

Regards,

Rudolf