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Divide polynomials over Galois field

`[quot,remd] = gfdeconv(b,a)`

[quot,remd] = gfdeconv(b,a,p)

[quot,remd] = gfdeconv(b,a,field)

This function performs computations in GF(p^{m}),
where p is prime. To work in GF(2^{m}), use
the `deconv`

function with Galois arrays. For details,
see Multiplication and Division of Polynomials.

The `gfdeconv`

function divides polynomials
over a Galois field. (To divide elements of a Galois field, use `gfdiv`

instead.) Algebraically, dividing
polynomials over a Galois field is equivalent to deconvolving vectors
containing the polynomials' coefficients, where the deconvolution
operation uses arithmetic over the same Galois field.

`[quot,remd] = gfdeconv(b,a)`

computes
the quotient `quot`

and remainder `remd`

of
the division of `b`

by `a`

in GF(2). `a`

and `b`

can
be either polynomial character vectors or
numeric vectors.

`[quot,remd] = gfdeconv(b,a,p)`

divides
the polynomial `b`

by the polynomial `a`

over
GF(`p`

) and returns the quotient in `quot`

and
the remainder in `remd`

. `p`

is
a prime number. `b`

, `a`

, `quot`

,
and `remd`

are row vectors that give the coefficients
of the corresponding polynomials in order of ascending powers. Each
coefficient is between 0 and `p`

-1.

`[quot,remd] = gfdeconv(b,a,field)`

divides
the polynomial `b`

by the polynomial `a`

over
GF(p^{m}) and returns the quotient in `quot`

and
the remainder in `remd`

. Here p is a prime number
and m is a positive integer. `b`

, `a`

, `quot`

,
and `remd`

are row vectors that list the exponential
formats of the coefficients of the corresponding polynomials, in order
of ascending powers. The exponential format is relative to some primitive
element of GF(p^{m}). `field`

is
the matrix listing all elements of GF(p^{m}),
arranged relative to the same primitive element. See Representing Elements of Galois Fields for
an explanation of these formats.

The code below shows that

$$(x+{x}^{3}+{x}^{4})\xf7(1+x)=1+{x}^{3}\text{Remainder}2$$

in GF(3). It also checks the results of the division.

p = 3; b = [0 1 0 1 1]; a = [1 1]; [quot, remd] = gfdeconv(b,a,p) % Check the result. bnew = gfadd(gfconv(quot,a,p),remd,p); if isequal(bnew,b) disp('Correct.') end;

The output is below.

quot = 1 0 0 1 remd = 2 Correct.

Working over GF(3), the code below outputs those polynomials
of the form x^{k} - 1
(k = 2, 3, 4,..., 8) that 1 + x^{2} divides
evenly.

p = 3; m = 2; a = [1 0 1]; % 1+x^2 for ii = 2:p^m-1 b = gfrepcov(ii); % x^ii b(1) = p-1; % -1+x^ii [quot, remd] = gfdeconv(b,a,p); % Display -1+x^ii if a divides it evenly. if remd==0 multiple{ii}=b; gfpretty(b) end end

The output is below.

4 2 + X 8 2 + X

In light of the discussion in Algorithms on the `gfprimck`

reference
page, along with the irreducibility of 1 + x^{2} over
GF(3), this output indicates that 1 + x^{2} is
not primitive for GF(9).

The algorithm of `gfdeconv`

is similar to that
of the MATLAB function `deconv`

.

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