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gdftmtx.m

## Copyright (C) 2002 David Bateman
##
## This program is free software; you can redistribute it and/or modify
## it under the terms of the GNU General Public License as published by
## the Free Software Foundation; either version 2 of the License, or
## (at your option) any later version.
##
## This program is distributed in the hope that it will be useful,
## but WITHOUT ANY WARRANTY; without even the implied warranty of
## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
## GNU General Public License for more details.
##
## You should have received a copy of the GNU General Public License
## along with this program; If not, see <http://www.gnu.org/licenses/>.

## -*- texinfo -*-
## @deftypefn {Function File} {@var{d} = } gdftmtx (@var{a})
##
## Form a matrix, that can be used to perform Fourier transforms in
## a Galois Field.
##
## Given that @var{a} is an element of the Galois Field GF(2^m), and
## that the minimum value for @var{k} for which @code{@var{a} ^ @var{k}} 
## is equal to one is @code{2^m - 1}, then this function produces a 
## @var{k}-by-@var{k} matrix representing the discrete Fourier transform    
## over a Galois Field with respect to @var{a}. The Fourier transform of
## a column vector is then given by @code{gdftmtx(@var{a}) * @var{x}}.
##
## The inverse Fourier transform is given by @code{gdftmtx(1/@var{a})}
## @end deftypefn
## @seealso{dftmtx}

## PKG_ADD: dispatch ("dftmtx", "gdftmtx", "galois");
function d = gdftmtx(a)

  if (nargin != 1)
    error ("usage: d = gdftmtx (a)");
  endif
    
  if (!isgalois(a))
    error("gdftmtx: argument must be a galois variable");
  endif

  m = a.m;
  prim = a.prim_poly;
  n = 2^a.m - 1;
  if (n > 255)
    error ([ "gdftmtx: argument must be in Galois Field GF(2^m), where" ...
           " m is not greater than 8"]); 
  endif

  if (length(a) ~= 1)
    error ("gdftmtx: argument must be a scalar");
  endif

  mp = minpol(a);
  if ((mp(1) ~= 1) | !isprimitive(mp))
    error("gdftmtx: argument must be a primitive nth root of unity");
  endif
  
  step = glog(a);
  step = step.x;
  row = gexp(gf([0:n-1], m, prim));
  d = zeros(n);
  for i=1:n;
    d(i,:) = row .^ mod(step*(i-1),n);
  end
  
endfunction

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