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classmethod galois.FieldArray.repr_table(element: ElementLike | None =
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: ElementLike | None =
None¶ An element to use as the exponent base in the power representation. The default is
Nonewhich corresponds toprimitive_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 ofelement. Sorting by any of the others will order the rows in lexicographically-increasing polynomial/vector order, which is equivalent to ascending order of the integer representation.
- element: ElementLike | None =
- Returns¶
A string representation of the table comparing the power, polynomial, vector, and integer representations of each field element.
Examples¶
Create a
FieldArraysubclass 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: xGenerate 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]: 26Generate 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]: 26Generate 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