classmethod galois.FieldArray.repr_table(element: = None, sort: 'power' | 'poly' | 'vector' | 'int' = 'power') str

Generates a finite field element representation table comparing the power, polynomial, vector, and integer representations.

Parameters:
element: = None

An element to use as the exponent base in the power representation. The default is None which corresponds to primitive_element.

sort: 'power' | 'poly' | 'vector' | 'int' = 'power'

The sorting method for the table. The default is "power". Sorting by "power" will order the rows of the table by ascending powers of element. Sorting by any of the others will order the rows in lexicographical polynomial/vector order, which is equivalent to ascending order of the integer representation.

Returns:

A string representation of the table comparing the power, polynomial, vector, and integer representations of each field element.

Examples

Create a FieldArray subclass for $$\mathrm{GF}(3^3)$$.

In [1]: GF = galois.GF(3**3)

In [2]: print(GF.properties)
Galois Field:
name: GF(3^3)
characteristic: 3
degree: 3
order: 27
irreducible_poly: x^3 + 2x + 1
is_primitive_poly: True
primitive_element: x


Generate a representation table for $$\mathrm{GF}(3^3)$$. Since $$x^3 + 2x + 1$$ is a primitive polynomial, $$x$$ is a primitive element of the field. Notice, $$\textrm{ord}(x) = 26$$.

In [3]: print(GF.repr_table())
Power     Polynomial      Vector    Integer
------- --------------- ----------- ---------
0           0         [0, 0, 0]      0
x^0          1         [0, 0, 1]      1
x^1          x         [0, 1, 0]      3
x^2         x^2        [1, 0, 0]      9
x^3        x + 2       [0, 1, 2]      5
x^4       x^2 + 2x     [1, 2, 0]      15
x^5     2x^2 + x + 2   [2, 1, 2]      23
x^6     x^2 + x + 1    [1, 1, 1]      13
x^7     x^2 + 2x + 2   [1, 2, 2]      17
x^8       2x^2 + 2     [2, 0, 2]      20
x^9        x + 1       [0, 1, 1]      4
x^10      x^2 + x      [1, 1, 0]      12
x^11    x^2 + x + 2    [1, 1, 2]      14
x^12      x^2 + 2      [1, 0, 2]      11
x^13         2         [0, 0, 2]      2
x^14         2x        [0, 2, 0]      6
x^15        2x^2       [2, 0, 0]      18
x^16       2x + 1      [0, 2, 1]      7
x^17      2x^2 + x     [2, 1, 0]      21
x^18    x^2 + 2x + 1   [1, 2, 1]      16
x^19   2x^2 + 2x + 2   [2, 2, 2]      26
x^20    2x^2 + x + 1   [2, 1, 1]      22
x^21      x^2 + 1      [1, 0, 1]      10
x^22       2x + 2      [0, 2, 2]      8
x^23     2x^2 + 2x     [2, 2, 0]      24
x^24   2x^2 + 2x + 1   [2, 2, 1]      25
x^25      2x^2 + 1     [2, 0, 1]      19

In [4]: GF("x").multiplicative_order()
Out[4]: 26


Generate a representation table for $$\mathrm{GF}(3^3)$$ using a different primitive element $$2x^2 + 2x + 2$$. Notice, $$\textrm{ord}(2x^2 + 2x + 2) = 26$$.

In [5]: print(GF.repr_table("2x^2 + 2x + 2"))
Power            Polynomial      Vector    Integer
-------------------- --------------- ----------- ---------
0                  0         [0, 0, 0]      0
(2x^2 + 2x + 2)^0          1         [0, 0, 1]      1
(2x^2 + 2x + 2)^1    2x^2 + 2x + 2   [2, 2, 2]      26
(2x^2 + 2x + 2)^2       x^2 + 2      [1, 0, 2]      11
(2x^2 + 2x + 2)^3     2x^2 + x + 2   [2, 1, 2]      23
(2x^2 + 2x + 2)^4    2x^2 + 2x + 1   [2, 2, 1]      25
(2x^2 + 2x + 2)^5       2x^2 + x     [2, 1, 0]      21
(2x^2 + 2x + 2)^6       x^2 + x      [1, 1, 0]      12
(2x^2 + 2x + 2)^7        x + 2       [0, 1, 2]      5
(2x^2 + 2x + 2)^8        2x + 2      [0, 2, 2]      8
(2x^2 + 2x + 2)^9         2x^2       [2, 0, 0]      18
(2x^2 + 2x + 2)^10      2x^2 + 2     [2, 0, 2]      20
(2x^2 + 2x + 2)^11         x         [0, 1, 0]      3
(2x^2 + 2x + 2)^12    2x^2 + x + 1   [2, 1, 1]      22
(2x^2 + 2x + 2)^13         2         [0, 0, 2]      2
(2x^2 + 2x + 2)^14    x^2 + x + 1    [1, 1, 1]      13
(2x^2 + 2x + 2)^15      2x^2 + 1     [2, 0, 1]      19
(2x^2 + 2x + 2)^16    x^2 + 2x + 1   [1, 2, 1]      16
(2x^2 + 2x + 2)^17    x^2 + x + 2    [1, 1, 2]      14
(2x^2 + 2x + 2)^18      x^2 + 2x     [1, 2, 0]      15
(2x^2 + 2x + 2)^19     2x^2 + 2x     [2, 2, 0]      24
(2x^2 + 2x + 2)^20       2x + 1      [0, 2, 1]      7
(2x^2 + 2x + 2)^21       x + 1       [0, 1, 1]      4
(2x^2 + 2x + 2)^22        x^2        [1, 0, 0]      9
(2x^2 + 2x + 2)^23      x^2 + 1      [1, 0, 1]      10
(2x^2 + 2x + 2)^24         2x        [0, 2, 0]      6
(2x^2 + 2x + 2)^25    x^2 + 2x + 2   [1, 2, 2]      17

In [6]: GF("2x^2 + 2x + 2").multiplicative_order()
Out[6]: 26


Generate a representation table for $$\mathrm{GF}(3^3)$$ using a non-primitive element $$x^2$$. Notice, $$\textrm{ord}(x^2) = 13 \ne 26$$.

In [7]: print(GF.repr_table("x^2"))
Power       Polynomial      Vector    Integer
---------- --------------- ----------- ---------
0             0         [0, 0, 0]      0
(x^2)^0          1         [0, 0, 1]      1
(x^2)^1         x^2        [1, 0, 0]      9
(x^2)^2       x^2 + 2x     [1, 2, 0]      15
(x^2)^3     x^2 + x + 1    [1, 1, 1]      13
(x^2)^4       2x^2 + 2     [2, 0, 2]      20
(x^2)^5       x^2 + x      [1, 1, 0]      12
(x^2)^6       x^2 + 2      [1, 0, 2]      11
(x^2)^7          2x        [0, 2, 0]      6
(x^2)^8        2x + 1      [0, 2, 1]      7
(x^2)^9     x^2 + 2x + 1   [1, 2, 1]      16
(x^2)^10    2x^2 + x + 1   [2, 1, 1]      22
(x^2)^11       2x + 2      [0, 2, 2]      8
(x^2)^12   2x^2 + 2x + 1   [2, 2, 1]      25
(x^2)^13         1         [0, 0, 1]      1
(x^2)^14        x^2        [1, 0, 0]      9
(x^2)^15      x^2 + 2x     [1, 2, 0]      15
(x^2)^16    x^2 + x + 1    [1, 1, 1]      13
(x^2)^17      2x^2 + 2     [2, 0, 2]      20
(x^2)^18      x^2 + x      [1, 1, 0]      12
(x^2)^19      x^2 + 2      [1, 0, 2]      11
(x^2)^20         2x        [0, 2, 0]      6
(x^2)^21       2x + 1      [0, 2, 1]      7
(x^2)^22    x^2 + 2x + 1   [1, 2, 1]      16
(x^2)^23    2x^2 + x + 1   [2, 1, 1]      22
(x^2)^24       2x + 2      [0, 2, 2]      8
(x^2)^25   2x^2 + 2x + 1   [2, 2, 1]      25

In [8]: GF("x^2").multiplicative_order()
Out[8]: 13