Communications Toolbox

gf

Create a Galois field array

Syntax

• x_gf = gf(x,m);
  x_gf = gf(x,m,prim_poly);
  x_gf = gf(x);
  

Description

x_gf = gf(x,m) creates a Galois field array from the matrix x. The Galois field has 2^m elements, where m is an integer between 1 and 16. The elements of x must be integers between 0 and 2^m-1. The output x_gf is a variable that MATLAB recognizes as a Galois field array, rather than an array of integers. As a result, when you manipulate x_gf using operators or functions such as + or det, MATLAB works within the Galois field you have specified.

Note    To learn how to manipulate x_gf using familiar MATLAB operators and functions, see Galois Field Computations. To learn how the integers in x represent elements of GF(2^m), see How Integers Correspond to Galois Field Elements.

x_gf = gf(x,m,prim_poly) is the same as the previous syntax, except that it uses the primitive polynomial prim_poly to define the field. prim_poly is the integer representation of a primitive polynomial. For example, the number 41 represents the polynomial D^5+D^2+1 because the binary form of 41 is 1 0 0 1 0 1. For more information about the primitive polynomial, see Specifying the Primitive Polynomial.

x_gf = gf(x) creates a GF(2) array from the matrix x. Each element of x must be 0 or 1.

Default Primitive Polynomials

The table below lists the primitive polynomial that gf uses by default for each Galois field GF(2^m). To use a different primitive polynomial, specify prim_poly as an input argument when you invoke gf.

m	Default Primitive Polynomial	Integer Representation
1	D + 1	3
2	D^2 + D + 1	7
3	D^3 + D + 1	11
4	D^4 + D + 1	19
5	D^5 + D^2 + 1	37
6	D^6 + D + 1	67
7	D^7 + D^3 + 1	137
8	D^8 + D^4 + D^3 + D^2 + 1	285
9	D^9 + D^4 + 1	529
10	D^10 + D^3 + 1	1033
11	D^11 + D^2 + 1	2053
12	D^12 + D^6 + D^4 + D + 1	4179
13	D^13 + D^4 + D^3 + D + 1	8219
14	D^14 + D^10 + D^6 + D + 1	17475
15	D^15 + D + 1	32771
16	D^16 + D^12 + D^3 + D + 1	69643

Examples

For examples that use gf, see

• Example: Creating Galois Field Variables
• Example: Representing a Primitive Element
• Other sample code within Galois Field Computations
• The Galois field demonstration: type playshow gfdemo.

See Also

gftable, list of functions and operators for Galois field computations, gfdemo

gen2par

gfadd