Galois Fields

The Java-Source

Base Classes

 GaloisField.java The base class for working with Galois Fields GF(p) (Documentation).
 ExtendedGaloisField.java The extended class for working with Extended Galois Fields GF(p^k) (Documentation).
 GaloisException.java  An Exception class (Documentation).
 GaloisPolynomial.java Polynomials with Galois coefficients (Documentation).

Code Base Classes

CyclicCode.java The base class for working with Galois Polynomial Codes (CycloDemoApplet) (Documentation).
BCHCode.java A special type of Cyclyc codes with parameters n, q, t and l.
RSCode.java A special type of BCH codes.

Demo Programs

 GaloisDemo.java Demo program for working with Galois Polynomials (AppletPreview).
 ExtendedDemo.java, ExtendedDemoApplet.java Demo program for working with Extended Galois Fields (Preview, AppletPreview).
 CycloDemoApplet Demo applet for cyclic codes.

Polynomials with Galois coefficients

 

Extended Galois Field construction

ExtendedDemo.java

Construction of GF(9)

Base : Base Galois Field GF(3).
Primitive Polynomial : 2 + a + a^2
Field : Extended Galois Field GF(9).
0 = 0
a^0 = 1
a^1 = a
a^2 = 1 + 2a
a^3 = 2 + 2a
a^4 = 2
a^5 = 2a
a^6 = 2 + a
a^7 = 1 + a

Isomorph fields GF(27)

Base : Base Galois Field GF(3).
Create Extended fields isomorph with GF(27) using irreducible polynomials
-------------------------------------------------------------------------
Polynomial: 1 + 2a + a^3 Alfa: a
Polynomial: 2 + 2a + a^3 Alfa: 2a
Polynomial: 2 + a^2 + a^3 Alfa: 2 + a
Polynomial: 2 + a + a^2 + a^3 Alfa: 1 + a
Polynomial: 1 + 2a + a^2 + a^3 Alfa: a
Polynomial: 1 + 2a^2 + a^3 Alfa: a
Polynomial: 1 + a + 2a^2 + a^3 Alfa: a
Polynomial: 2 + 2a + 2a^2 + a^3 Alfa: 1 + a

Extended fields GF(2^k)

Base : Base Galois Field GF(2).
Create Extended fields GF(2^k). Show ModuloPoly and Primitive Element
---------------------------------------------------------------------
Extended Galois Field GF(4). 1 + a + a^2, a
Extended Galois Field GF(8). 1 + a + a^3, a
Extended Galois Field GF(16). 1 + a + a^4, a
Extended Galois Field GF(32). 1 + a^2 + a^5, a
Extended Galois Field GF(64). 1 + a + a^6, a
Extended Galois Field GF(128). 1 + a + a^7, a

Minimal polynomials

Base: GF(3), Extended Galois Field GF(9).
Cyclotomics modulo 8, using GF(3)
---------------------------------
{0}
           2 + x
{1, 3}
           2 + x + x^2
{2, 6}
           1 + x^2
{4}
           1 + x
{5, 7}
           2 + 2x + x^2

Cyclyc Code Demo

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