Description
The following is a list of
primitive irreducible polynomials
for generating elements of a binary extension field GF(2m )
from a base finite field. The list contains polynomials of degree 2 to 32.
Downloads
x^2 + x^1 + 1
x^3 + x^1 + 1
x^4 + x^1 + 1
x^5 + x^2 + 1
x^5 + x^4 + x^2 + x^1 + 1
x^5 + x^4 + x^3 + x^2 + 1
x^6 + x^1 + 1
x^6 + x^5 + x^2 + x^1 + 1
x^6 + x^5 + x^3 + x^2 + 1
x^7 + x^1 + 1
x^7 + x^3 + 1
x^7 + x^3 + x^2 + x^1 + 1
x^7 + x^4 + x^3 + x^2 + 1
x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^7 + x^6 + x^3 + x^1 + 1
x^7 + x^6 + x^4 + x^2 + 1
x^7 + x^6 + x^5 + x^2 + 1
x^7 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^8 + x^4 + x^3 + x^2 + 1
x^8 + x^5 + x^3 + x^1 + 1
x^8 + x^6 + x^4 + x^3 + x^2 + x^1 + 1
x^8 + x^6 + x^5 + x^1 + 1
x^8 + x^6 + x^5 + x^2 + 1
x^8 + x^6 + x^5 + x^3 + 1
x^8 + x^7 + x^6 + x^1 + 1
x^8 + x^7 + x^6 + x^5 + x^2 + x^1 + 1
x^9 + x^4 + 1
x^9 + x^5 + x^3 + x^2 + 1
x^9 + x^6 + x^4 + x^3 + 1
x^9 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^9 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1
x^9 + x^8 + x^4 + x^1 + 1
x^9 + x^8 + x^5 + x^4 + 1
x^9 + x^8 + x^6 + x^5 + 1
x^9 + x^8 + x^6 + x^5 + x^3 + x^1 + 1
x^9 + x^8 + x^7 + x^2 + 1
x^9 + x^8 + x^7 + x^3 + x^2 + x^1 + 1
x^9 + x^8 + x^7 + x^6 + x^5 + x^1 + 1
x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + 1
x^10 + x^3 + 1
x^10 + x^4 + x^3 + x^1 + 1
x^10 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^10 + x^8 + x^3 + x^2 + 1
x^10 + x^8 + x^4 + x^3 + 1
x^10 + x^8 + x^5 + x^1 + 1
x^10 + x^8 + x^5 + x^4 + 1
x^10 + x^8 + x^7 + x^6 + x^5 + x^2 + 1
x^10 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^1 + 1
x^10 + x^9 + x^4 + x^1 + 1
x^10 + x^9 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^10 + x^9 + x^8 + x^6 + x^3 + x^2 + 1
x^10 + x^9 + x^8 + x^6 + x^5 + x^1 + 1
x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + 1
x^11 + x^2 + 1
x^11 + x^5 + x^3 + x^1 + 1
x^11 + x^5 + x^3 + x^2 + 1
x^11 + x^6 + x^5 + x^1 + 1
x^11 + x^7 + x^3 + x^2 + 1
x^11 + x^8 + x^5 + x^2 + 1
x^11 + x^8 + x^6 + x^5 + x^4 + x^1 + 1
x^11 + x^8 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^11 + x^9 + x^4 + x^1 + 1
x^11 + x^9 + x^8 + x^7 + x^4 + x^1 + 1
x^11 + x^10 + x^3 + x^2 + 1
x^11 + x^10 + x^7 + x^4 + x^3 + x^1 + 1
x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^1 + 1
x^11 + x^10 + x^9 + x^8 + x^3 + x^1 + 1
x^12 + x^6 + x^4 + x^1 + 1
x^12 + x^9 + x^3 + x^2 + 1
x^12 + x^9 + x^8 + x^3 + x^2 + x^1 + 1
x^12 + x^10 + x^9 + x^8 + x^6 + x^2 + 1
x^12 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + 1
x^12 + x^11 + x^6 + x^4 + x^2 + x^1 + 1
x^12 + x^11 + x^9 + x^5 + x^3 + x^1 + 1
x^12 + x^11 + x^9 + x^7 + x^6 + x^4 + 1
x^12 + x^11 + x^9 + x^7 + x^6 + x^5 + 1
x^12 + x^11 + x^9 + x^8 + x^7 + x^4 + 1
x^12 + x^11 + x^9 + x^8 + x^7 + x^5 + x^2 + x^1 + 1
x^12 + x^11 + x^10 + x^5 + x^2 + x^1 + 1
x^12 + x^11 + x^10 + x^8 + x^6 + x^4 + x^3 + x^1 + 1
x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^5 + x^4 + x^3 + x^1 + 1
x^13 + x^4 + x^3 + x^1 + 1
x^13 + x^9 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^13 + x^9 + x^8 + x^7 + x^5 + x^1 + 1
x^13 + x^10 + x^9 + x^7 + x^5 + x^4 + 1
x^13 + x^10 + x^9 + x^8 + x^6 + x^3 + x^2 + x^1 + 1
x^13 + x^11 + x^8 + x^7 + x^4 + x^1 + 1
x^13 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^13 + x^12 + x^6 + x^5 + x^4 + x^3 + 1
x^13 + x^12 + x^8 + x^7 + x^6 + x^5 + 1
x^13 + x^12 + x^9 + x^8 + x^4 + x^2 + 1
x^13 + x^12 + x^10 + x^8 + x^6 + x^4 + x^3 + x^2 + 1
x^13 + x^12 + x^11 + x^5 + x^2 + x^1 + 1
x^13 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^1 + 1
x^13 + x^12 + x^11 + x^9 + x^5 + x^3 + 1
x^14 + x^8 + x^6 + x^1 + 1
x^14 + x^10 + x^6 + x^1 + 1
x^14 + x^10 + x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1
x^14 + x^11 + x^6 + x^1 + 1
x^14 + x^11 + x^9 + x^6 + x^5 + x^2 + 1
x^14 + x^12 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + 1
x^14 + x^12 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^1 + 1
x^14 + x^12 + x^11 + x^10 + x^9 + x^7 + x^4 + x^3 + 1
x^14 + x^13 + x^6 + x^5 + x^3 + x^1 + 1
x^14 + x^13 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^14 + x^13 + x^11 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^14 + x^13 + x^11 + x^8 + x^5 + x^3 + x^2 + x^1 + 1
x^14 + x^13 + x^12 + x^11 + x^10 + x^7 + x^6 + x^1 + 1
x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^6 + x^5 + 1
x^15 + x^1 + 1
x^15 + x^4 + 1
x^15 + x^7 + 1
x^15 + x^7 + x^6 + x^3 + x^2 + x^1 + 1
x^15 + x^10 + x^5 + x^1 + 1
x^15 + x^10 + x^5 + x^4 + 1
x^15 + x^10 + x^5 + x^4 + x^2 + x^1 + 1
x^15 + x^10 + x^9 + x^7 + x^5 + x^3 + 1
x^15 + x^10 + x^9 + x^8 + x^5 + x^3 + 1
x^15 + x^11 + x^7 + x^6 + x^2 + x^1 + 1
x^15 + x^12 + x^3 + x^1 + 1
x^15 + x^12 + x^5 + x^4 + x^3 + x^2 + 1
x^15 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^2 + 1
x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
x^16 + x^9 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + 1
x^16 + x^12 + x^3 + x^1 + 1
x^16 + x^12 + x^7 + x^2 + 1
x^16 + x^13 + x^12 + x^10 + x^9 + x^7 + x^6 + x^1 + 1
x^16 + x^13 + x^12 + x^11 + x^7 + x^6 + x^3 + x^1 + 1
x^16 + x^13 + x^12 + x^11 + x^10 + x^6 + x^2 + x^1 + 1
x^16 + x^14 + x^10 + x^8 + x^3 + x^1 + 1
x^16 + x^14 + x^13 + x^12 + x^6 + x^5 + x^3 + x^2 + 1
x^16 + x^14 + x^13 + x^12 + x^10 + x^7 + 1
x^16 + x^15 + x^10 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^10 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^10 + x^9 + x^6 + x^2 + x^1 + 1
x^16 + x^15 + x^11 + x^10 + x^9 + x^8 + x^6 + x^4 + x^2 + x^1 + 1
x^17 + x^3 + 1
x^17 + x^3 + x^2 + x^1 + 1
x^17 + x^5 + 1
x^17 + x^6 + 1
x^17 + x^8 + x^4 + x^3 + 1
x^17 + x^8 + x^7 + x^6 + x^4 + x^3 + 1
x^17 + x^10 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + 1
x^17 + x^12 + x^6 + x^3 + x^2 + x^1 + 1
x^17 + x^12 + x^9 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^17 + x^12 + x^9 + x^7 + x^6 + x^4 + x^3 + x^2 + 1
x^17 + x^14 + x^11 + x^7 + x^5 + x^3 + x^2 + x^1 + 1
x^17 + x^15 + x^13 + x^11 + x^9 + x^7 + x^5 + x^3 + 1
x^17 + x^15 + x^13 + x^11 + x^9 + x^7 + x^6 + x^4 + x^2 + x^1 + 1
x^17 + x^16 + x^3 + x^1 + 1
x^18 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^18 + x^7 + 1
x^18 + x^7 + x^5 + x^2 + x^1 + 1
x^18 + x^8 + x^2 + x^1 + 1
x^18 + x^9 + x^7 + x^6 + x^5 + x^4 + 1
x^18 + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + x^1 + 1
x^18 + x^9 + x^8 + x^7 + x^6 + x^4 + x^2 + x^1 + 1
x^18 + x^10 + x^7 + x^5 + 1
x^18 + x^10 + x^8 + x^5 + 1
x^18 + x^10 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^18 + x^10 + x^9 + x^3 + 1
x^18 + x^13 + x^6 + x^4 + 1
x^18 + x^15 + x^5 + x^2 + 1
x^18 + x^15 + x^9 + x^2 + 1
x^19 + x^5 + x^2 + x^1 + 1
x^19 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^6 + x^2 + x^1 + 1
x^19 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^19 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
x^19 + x^7 + x^5 + x^3 + x^2 + x^1 + 1
x^19 + x^8 + x^7 + x^5 + 1
x^19 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^8 + x^7 + x^6 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^9 + x^8 + x^5 + 1
x^19 + x^9 + x^8 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^19 + x^9 + x^8 + x^7 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + 1
x^19 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^3 + x^2 + x^1 + 1
x^19 + x^16 + x^13 + x^10 + x^7 + x^4 + x^1 + 1
x^20 + x^3 + 1
x^20 + x^9 + x^5 + x^3 + 1
x^20 + x^11 + x^8 + x^6 + x^3 + x^2 + 1
x^20 + x^14 + x^10 + x^9 + x^8 + x^6 + x^5 + x^4 + 1
x^20 + x^17 + x^14 + x^10 + x^7 + x^4 + x^3 + x^2 + 1
x^20 + x^19 + x^4 + x^3 + 1
x^21 + x^2 + 1
x^21 + x^8 + x^7 + x^4 + x^3 + x^2 + 1
x^21 + x^10 + x^6 + x^4 + x^3 + x^2 + 1
x^21 + x^13 + x^5 + x^2 + 1
x^21 + x^14 + x^7 + x^2 + 1
x^21 + x^14 + x^7 + x^6 + x^3 + x^2 + 1
x^21 + x^14 + x^12 + x^7 + x^6 + x^4 + x^3 + x^2 + 1
x^21 + x^15 + x^10 + x^9 + x^5 + x^4 + x^3 + x^2 + 1
x^21 + x^20 + x^19 + x^18 + x^5 + x^4 + x^3 + x^2 + 1
x^22 + x^1 + 1
x^22 + x^9 + x^5 + x^1 + 1
x^22 + x^14 + x^13 + x^12 + x^7 + x^3 + x^2 + x^1 + 1
x^22 + x^17 + x^9 + x^7 + x^2 + x^1 + 1
x^22 + x^17 + x^13 + x^12 + x^8 + x^7 + x^2 + x^1 + 1
x^22 + x^20 + x^18 + x^16 + x^6 + x^4 + x^2 + x^1 + 1
x^23 + x^5 + 1
x^23 + x^5 + x^4 + x^1 + 1
x^23 + x^11 + x^10 + x^7 + x^6 + x^5 + 1
x^23 + x^12 + x^5 + x^4 + 1
x^23 + x^15 + x^10 + x^9 + x^7 + x^5 + x^4 + x^3 + 1
x^23 + x^16 + x^13 + x^6 + x^5 + x^3 + 1
x^23 + x^17 + x^11 + x^5 + 1
x^23 + x^17 + x^11 + x^9 + x^8 + x^5 + x^4 + x^1 + 1
x^23 + x^18 + x^16 + x^13 + x^11 + x^8 + x^5 + x^2 + 1
x^23 + x^21 + x^7 + x^5 + 1
x^24 + x^7 + x^2 + x^1 + 1
x^24 + x^21 + x^19 + x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^10 + x^9 + x^5 + x^4 + x^1 + 1
x^24 + x^22 + x^20 + x^18 + x^16 + x^14 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + 1
x^25 + x^3 + 1
x^25 + x^3 + x^2 + x^1 + 1
x^25 + x^11 + x^9 + x^8 + x^6 + x^4 + x^3 + x^2 + 1
x^25 + x^12 + x^4 + x^3 + 1
x^25 + x^12 + x^11 + x^8 + x^7 + x^6 + x^4 + x^3 + 1
x^25 + x^17 + x^10 + x^3 + x^2 + x^1 + 1
x^25 + x^18 + x^12 + x^11 + x^6 + x^5 + x^4 + x^3 + 1
x^25 + x^20 + x^5 + x^3 + 1
x^25 + x^20 + x^16 + x^11 + x^5 + x^3 + x^2 + x^1 + 1
x^25 + x^23 + x^21 + x^19 + x^9 + x^7 + x^5 + x^3 + 1
x^26 + x^6 + x^2 + x^1 + 1
x^26 + x^19 + x^16 + x^15 + x^14 + x^13 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + 1
x^26 + x^21 + x^18 + x^16 + x^15 + x^13 + x^12 + x^11 + x^9 + x^8 + x^6 + x^5 + x^4 + x^3 + 1
x^26 + x^22 + x^20 + x^19 + x^16 + x^13 + x^11 + x^9 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1
x^26 + x^22 + x^21 + x^16 + x^12 + x^11 + x^10 + x^8 + x^5 + x^4 + x^3 + x^1 + 1
x^26 + x^23 + x^22 + x^21 + x^19 + x^18 + x^15 + x^14 + x^13 + x^11 + x^10 + x^9 + x^8 +x^6+ x^5+x^2+1
x^26 + x^24 + x^21 + x^17 + x^16 + x^14 + x^13 + x^11 + x^7 + x^6 + x^4 + x^1 + 1
x^27 + x^5 + x^2 + x^1 + 1
x^27 + x^18 + x^11 + x^10 + x^9 + x^5 + x^4 + x^3 + 1
x^27 + x^22 + x^13 + x^11 + x^6 + x^5 + x^4 + x^3 + 1
x^27 + x^22 + x^17 + x^15 + x^14 + x^13 + x^6 + x^1 + 1
x^27 + x^22 + x^21 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 + x^7 + x^5 + 1
x^27 + x^24 + x^19 + x^16 + x^12 + x^8 + x^7 + x^3 + x^2 + x^1 + 1
x^27 + x^24 + x^21 + x^19 + x^16 + x^13 + x^11 + x^9 + x^6 + x^5 + x^4 + x^3 + 1
x^27 + x^25 + x^23 + x^21 + x^13 + x^11 + x^9 + x^8 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^27 + x^25 + x^23 + x^21 + x^20 + x^19 + x^18 + x^16 + x^14 + x^10 + x^8 + x^7 + x^4 + x^3 + 1
x^28 + x^3 + 1
x^28 + x^13 + x^11 + x^9 + x^5 + x^3 + 1
x^28 + x^18 + x^17 + x^16 + x^9 + x^5 + x^4 + x^3 + 1
x^28 + x^19 + x^17 + x^15 + x^10 + x^6 + x^3 + x^2 + 1
x^28 + x^22 + x^11 + x^10 + x^4 + x^3 + 1
x^28 + x^24 + x^20 + x^16 + x^12 + x^8 + x^4 + x^3 + 1
x^29 + x^2 + 1
x^29 + x^12 + x^7 + x^2 + 1
x^29 + x^18 + x^14 + x^6 + x^3 + x^2 + 1
x^29 + x^19 + x^16 + x^6 + x^3 + x^2 + 1
x^29 + x^20 + x^11 + x^2 + 1
x^29 + x^20 + x^16 + x^11 + x^8 + x^4 + x^3 + x^2 + 1
x^29 + x^21 + x^5 + x^2 + 1
x^29 + x^23 + x^10 + x^9 + x^5 + x^4 + x^3 + x^2 + 1
x^29 + x^24 + x^14 + x^13 + x^8 + x^4 + x^3 + x^2 + 1
x^29 + x^26 + x^5 + x^2 + 1
x^30 + x^23 + x^2 + x^1 + 1
x^30 + x^24 + x^20 + x^16 + x^14 + x^13 + x^11 + x^7 + x^2 + x^1 + 1
x^30 + x^24 + x^21 + x^20 + x^18 + x^15 + x^13 + x^12 + x^9 + x^7 + x^6 + x^4 + x^3 + x^1 + 1
x^30 + x^25 + x^24 + x^23 + x^19 + x^18 + x^16 + x^14 + x^11 + x^8 + x^6 + x^4 + x^3 + x^1 + 1
x^30 + x^27 + x^25 + x^24 + x^23 + x^22 + x^19 + x^16 + x^12 + x^10 + x^8 + x^7 + x^6 + x^1 + 1
x^31 + x^3 + 1
x^31 + x^3 + x^2 + x^1 + 1
x^31 + x^13 + x^8 + x^3 + 1
x^31 + x^16 + x^8 + x^4 + x^3 + x^2 + 1
x^31 + x^20 + x^15 + x^5 + x^4 + x^3 + 1
x^31 + x^20 + x^18 + x^7 + x^5 + x^3 + 1
x^31 + x^21 + x^12 + x^3 + x^2 + x^1 + 1
x^31 + x^23 + x^22 + x^15 + x^14 + x^7 + x^4 + x^3 + 1
x^31 + x^25 + x^19 + x^14 + x^7 + x^3 + x^2 + x^1 + 1
x^31 + x^27 + x^23 + x^19 + x^15 + x^11 + x^7 + x^3 + 1
x^31 + x^27 + x^23 + x^19 + x^15 + x^11 + x^10 + x^9 + x^7 + x^6 + x^5 + x^3 + x^2 + x^1 + 1
x^32 + x^22 + x^2 + x^1 + 1
x^32 + x^22 + x^21 + x^20 + x^18 + x^17 + x^15 + x^13 + x^12 + x^10 + x^8 + x^6 + x^4 + x^1 + 1
x^32 + x^23 + x^17 + x^16 + x^14 + x^10 + x^8 + x^7 + x^6 + x^5 + x^3 + 1
x^32 + x^26 + x^23 + x^22 + x^16 + x^12 + x^11 + x^10 + x^8 + x^7 + x^5 + x^4 + x^2 + x^1 + 1
x^32 + x^27 + x^26 + x^25 + x^24 + x^23 + x^22 + x^17 + x^13 + x^11 + x^10 + x^9 +x^8 +x^7+x^2 +x^1+1
x^32 + x^28 + x^19 + x^18 + x^16 + x^14 + x^11 + x^10 + x^9 + x^6 + x^5 + x^1 + 1
Credits
Coherent Spread Spectrum Systems - J.K. Holmes Digital Communications And Spread Spectrum Systems - Rodger E. Ziemer and Roger L. Peterson Error Control Coding - S. Lin, D. Costello Error Correction Codes - W.W. Peterson And E.J. Weldon On Primitive Trinomials - N. Zierler and J. Brillhart Primitive Polynomials (Mod 2) - By E.J. Watson Sequence A011260/M0107 - N. J. A. Sloane Tables of Irreducible Polynomials For The First Four Prime Moduli - R. Church Primitive Polynomial Generation Algorithms Implementation And Performance Analysis
primitive_polynomials_GF2.txt