# Documentation

## Syntax

`c = gfadd(a,b) c = gfadd(a,b,p) c = gfadd(a,b,p,len) c = gfadd(a,b,field) `

## Description

 Note:   This function performs computations in GF(pm) where p is prime. To work in GF(2m), apply the + operator to Galois arrays of equal size. For details, see Example: Addition and Subtraction.

`c = gfadd(a,b) ` adds two GF(2) polynomials, `a` and `b`, which can be either polynomial strings or numeric vectors. If `a` and `b` are vectors of the same orientation but different lengths, then the shorter vector is zero-padded. If `a` and `b` are matrices they must be of the same size.

`c = gfadd(a,b,p) ` adds two GF(`p`) polynomials, where `p` is a prime number. `a`, `b`, and `c` are row vectors that give the coefficients of the corresponding polynomials in order of ascending powers. Each coefficient is between 0 and `p`-1. If `a` and `b` are matrices of the same size, the function treats each row independently.

`c = gfadd(a,b,p,len) ` adds row vectors `a` and `b` as in the previous syntax, except that it returns a row vector of length `len`. The output `c` is a truncated or extended representation of the sum. If the row vector corresponding to the sum has fewer than `len` entries (including zeros), extra zeros are added at the end; if it has more than `len` entries, entries from the end are removed.

`c = gfadd(a,b,field) ` adds two GF(pm) elements, where m is a positive integer. `a` and `b` are the exponential format of the two elements, 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. `c` is the exponential format of the sum, relative to the same primitive element. See Representing Elements of Galois Fields for an explanation of these formats. If `a` and `b` are matrices of the same size, the function treats each element independently.

## Examples

In the code below, `sum5` is the sum of 2 + 3x + x2 and 4 + 2x + 3x2 over GF(5), and `linpart` is the degree-one part of `sum5`.

```sum5 = gfadd([2 3 1],[4 2 3],5) linpart = gfadd([2 3 1],[4 2 3],5,2)```

The output is

```sum5 = 1 0 4 linpart = 1 0 ```

The code below shows that A2 + A4 = A1, where A is a root of the primitive polynomial 2 + 2x + x2 for GF(9).

```p = 3; m = 2; prim_poly = [2 2 1]; field = gftuple([-1:p^m-2]',prim_poly,p); g = gfadd(2,4,field)```

The output is

```g = 1 ```

Other examples are in Arithmetic in Galois Fields.