Communications Toolbox

Minimal Polynomials

The minimal polynomial of an element of GF(2^m) is the smallest-degree nonzero binary-coefficient polynomial having that element as a root in GF(2^m). To find the minimal polynomial of an element or a column vector of elements, use the minpol function.

The code below finds that the minimal polynomial of gf(6,4) is D² + D + 1 and then checks that gf(6,4) is indeed among the roots of that polynomial in the field GF(16).

m = 4;
e = gf(6,4);
em = minpol(e) % Find minimal polynomial of e. em is in GF(2).

emr = roots(gf([0 0 1 1 1],m)) % Roots of D^2+D+1 in GF(2^m)

The output is

em = GF(2) array. 
 
Array elements = 
 
     0     0     1     1     1


emr = GF(2^4) array. Primitive polynomial = D^4+D+1 (19 decimal)
 
Array elements = 
 
     6
     7

To find out which elements of a Galois field share the same minimal polynomial, use the cosets function.

Roots of Binary Polynomials

Manipulating Galois Variables

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