Primitive Polynomials

Source Notebook

Primitive polynomials for Galois field generation up to GF(2^1200), GF(3^660), GF(5^430), and GF(7^358)

(2644 elements)

Examples

Basic Examples

Retrieve the default content:

In[1]:=

Out[1]=

Visual Examples

Show the data for GF(2⁶⁷). The number 2⁶⁷-1 has a factorization 193707721×761838257287 (Cole, 1903) that gives a normalization close to 1:

In[2]:=

Out[2]=

Generate the first 600 powers of x using a polynomial modulus of 1+x+x²+x⁵+x⁶⁷:

In[3]:=

$With[{data = ResourceData["Primitive Polynomials"]}, ArrayPlot[ Transpose[ Table[PadRight[ CoefficientList[ PolynomialMod[\[FormalX]^n, {data[{2, 67}]["Polynomial"], 2}], \[FormalX]], 67, 0], {n, 1, 600}]], Frame -> False, ImageSize -> {610, 70}, PixelConstrained -> True]]$

Out[3]=

Show the approximate count via multiplication of the count normalization by (pⁿ-1)/n and compare that to the actual polynomial count:

In[4]:=

$Grid[{Row[{ NumberForm[#[[2]]["CountNormalization"], {7, 6}, NumberPadding -> {"", "0"}], "\[Times](", Superscript[#[[1, 1]], #[[1, 2]]], "-1)/", #[[1, 2]], " = ", Round[#[[2]][ "CountNormalization"] (#[[1, 1]]^#[[1, 2]] - 1)/#[[1, 2]]] }], Row[{ "exact count = ", #[[2]]["PolynomialCount"]}]} & /@ SortBy[Select[Normal[ResourceData["Primitive Polynomials"]], 10^15 < #[[1, 1]]^#[[1, 2]] < 10^16 &], #[[2]][ "PolynomialCount"] &], Frame -> All]$