For now, COS only generates polynomials over GF(2), GF(3), GF(4) and GF(5).
For GF(2), the coefficients are either 0 or 1 and the rules for multiplication and addition are 0 = 0+0 = 1+1 = 0*0 = 0*1 = 1*0 and 1 = 0+1 = 1+0 = 1*1.
A polynomial over GF(2) is irreducible if it cannot be factored into nontrivial polynomials. For example, x^{2}+x+1 is irreducible, but x^{2}+1 is not, since x^{2}+1 = (x+1)(x+1).
The order of a polynomial f(x) for which f(0) is not 0 is the smallest integer e for which f(x) divides x^{e}+1. A polynomial over GF(2) is primitive if it has order 2^{n}1. For example, x^{2}+x+1 has order 3 = 2^{2}1 since (x^{2}+x+1)(x+1) = x^{3}+1. Thus x^{2}+x+1 is primitive.
The number of degree n irreducible polynomials over GF(q) is
L_{q}(n) = 

 µ(n/d) q ^{d} , 
The number, L_{q}(n), of irreducible polynomials over GF(2) for n=1,2,...,15 is 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161, 2182. This is sequence A001037(M0116) in Neil J. Sloane's database of integer sequences.
The number of degree n primitive polynomials over GF(q) is Ø(q^{n} 1)/n, where Ø(m) is Euler's totient function, the number of numbers less than m and relatively prime to m. For example, Ø(63) = {2,4,5,8,...,62} = 36 and thus the number of primitive polynomials of degree 6 over GF(2) is 36/6 = 6.
The number of primitive polynomials over GF(2) for n=1,2,...,15 is 1, 1, 2, 2, 6, 6, 18, 16, 48, 60, 176, 144, 630, 756, 1800. This is sequence A011260(M0107) in Neil J. Sloane's database of integer sequences.
Below we show the output of COS for n=6 and irreducible polynomials. Primitive polynomials are output in blue. The meanings of the different output options should be obvious.
Output 


String 
Coefficients 
Polynomials 
1 0 0 0 0 1 1  6, 1, 0  x^{6} + x^{1} + 1 
1 0 1 0 1 1 1  6, 4, 2, 1, 0  x^{6} + x^{4} + x^{2} + x^{1} + 1 
1 1 0 0 1 1 1  6, 5, 2, 1, 0  x^{6} + x^{5} + x^{2} + x^{1} + 1 
1 0 0 1 0 0 1  6, 3, 0  x^{6} + x^{3} + 1 
1 1 0 1 1 0 1  6, 5, 3, 2, 0  x^{6} + x^{5} + x^{3} + x^{2} + 1 
1 0 1 1 0 1 1  6, 4, 3, 1, 0  x^{6} + x^{4} + x^{3} + x^{1} + 1 
1 1 1 0 1 0 1  6, 5, 4, 2, 0  x^{6} + x^{5} + x^{4} + x^{2} + 1 
1 1 1 0 0 1 1  6, 5, 4, 1, 0  x^{6} + x^{5} + x^{4} + x^{1} + 1 
1 1 0 0 0 0 1  6, 5, 0  x^{6} + x^{5} + 1 
Using our program we have computed some tables that may be of interest. The density of a polynomial is the number of nonzero terms. The index of a polynomial is the sum of the exponents of the nonzero terms.