# Documentation

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# gfdeconv

Divide polynomials over Galois field

## Syntax

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

## Description

### Note

This function performs computations in GF(pm), where p is prime. To work in GF(2m), 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(pm) 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(pm). `field` is the matrix listing all elements of GF(pm), arranged relative to the same primitive element. See Representing Elements of Galois Fields for an explanation of these formats.

## Examples

The code below shows that

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 xk - 1 (k = 2, 3, 4,..., 8) that 1 + x2 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 + x2 over GF(3), this output indicates that 1 + x2 is not primitive for GF(9).

## Algorithms

The algorithm of `gfdeconv` is similar to that of the MATLAB function `deconv`.