Communications Toolbox

Galois Field Computations

A Galois field is an algebraic field that has a finite number of members. This section describes how to work with fields that have 2^m members, where m is an integer between 1 and 16. Such fields are denoted GF(2^m). Galois fields having 2^m members are used in error-control coding. If you need to use Galois fields having an odd number of elements, see Appendix: Galois Fields of Odd Characteristic in the online documentation for the Communications Toolbox.

Galois Field Features of the Toolbox

The Communications Toolbox facilitates computations in Galois fields that have 2^m members. You can create array variables whose values are in GF(2^m) and use these variables to perform computations in the Galois field. Most computations use the same syntax that you would use to manipulate ordinary MATLAB arrays of real numbers, making the GF(2^m) capabilities of the toolbox easy to learn and use.

The topics in this section are

• Galois Field Terminology
• Representing Elements of Galois Fields
• Primitive Polynomials and Element Representations
• Arithmetic in Galois Fields
• Logical Operations in Galois Fields
• Matrix Manipulation in Galois Fields
• Linear Algebra in Galois Fields
• Signal Processing Operations in Galois Fields
• Polynomials over Galois Fields
• Manipulating Galois Variables
• Speed and Nondefault Primitive Polynomials

For background information about Galois fields or their use in error-control coding, see the works listed in Selected Bibliography for Galois Fields.

For more details about specific functions that process arrays of Galois field elements, see the online reference entries in the documentation for MATLAB or for the Communications Toolbox. Functions whose behavior is identical to the corresponding MATLAB function, except for the ability to handle Galois field members, do not have reference entries in this manual because the entries would be identical to those in the MATLAB manual.

Selected Bibliography for Special Filters

Galois Field Terminology