% -*- texinfo -*-
% @deftypefn {Function File} {@var{g} = } rsgenpoly (@var{n},@var{k})
% @deftypefnx {Function File} {@var{g} = } rsgenpoly (@var{n},@var{k},@var{p})
% @deftypefnx {Function File} {@var{g} = } rsgenpoly (@var{n},@var{k},@var{p},@var{b},@var{s})
% @deftypefnx {Function File} {@var{g} = } rsgenpoly (@var{n},@var{k},@var{p},@var{b})
% @deftypefnx {Function File} {[@var{g}, @var{t}] = } rsgenpoly (@var{...})
%
% Creates a generator polynomial for a Reed-Solomon coding with message
% length of @var{k} and codelength of @var{n}. @var{n} must be greater
% than @var{k} and their difference must be even. The generator polynomial
% is returned on @var{g} as a polynomial over the Galois Field GF(2^@var{m})
% where @var{n} is equal to @code{2^@var{m}-1}. If @var{m} is not integer
% the next highest integer value is used and a generator for a shorten
% Reed-Solomon code is returned.
%
% The elements of @var{g} represent the coefficients of the polynomial in
% descending order. If the length of @var{g} is lg, then the generator
% polynomial is given by
% @iftex
% @tex
% $$
% g_0 x^{lg-1} + g_1 x^{lg-2} + \cdots + g_{lg-1} x + g_lg.
% $$
% @end tex
% @end iftex
% @ifinfo
%
% @example
% @var{g}(0) * x^(lg-1) + @var{g}(1) * x^(lg-2) + ... + @var{g}(lg-1) * x + @var{g}(lg).
% @end example
% @end ifinfo
%
% If @var{p} is defined then it is used as the primitive polynomial of the
% the Galois Field GF(2^@var{m}). The default primitive polynomial will
% be used if @var{p} is equal to [].
%
% The variables @var{b} and @var{s} determine the form of the generator
% polynomial in the following manner.
% @iftex
% @tex
% $$
% g = (x - A^{bs}) (x - A^{(b+1)s}) \cdots (x - A ^{(b+2t-1)s}).
% $$
% @end tex
% @end iftex
% @ifinfo
%
% @example
% @var{g} = (@var{x} - A^(@var{b}*@var{s})) * (@var{x} - A^((@var{b}+1)*@var{s})) * ... * (@var{x} - A^((@var{b}+2*@var{t}-1)*@var{s})).
% @end example
% @end ifinfo
%
% where @var{t} is @code{(@var{n}-@var{k})/2}, and A is the primitive element
% of the Galois Field. Therefore @var{b} is the first consecutive root of the
% generator polynomial and @var{s} is the primitive element to generate the
% the polynomial roots.
%
% If requested the variable @var{t}, which gives the error correction
% capability of the the Reed-Solomon code
% @end deftypefn
% @seealso{gf,rsenc,rsdec}