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Create Galois field array
x_gf = gf(x,m)
x_gf = gf(x,m,prim_poly)
x_gf = gf(x)
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.
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 |
For examples that use gf, see
Other sample code within Galois Field Computations
The Galois field demonstration: type playshow gfdemo.
gftable, list of functions and operators for Galois field computations, gfdemo, Galois Field Computations
| | genqammod | gfadd | |
© 1994-2005 The MathWorks, Inc.