Elements of $GF(5^n)$ with given trace and subtrace
If $a$ is an element of $GF(5^n)$, then the (absolute)
trace of $a$ is
\[
\operatorname{tr}(a)=a+a^5+a^{25}+\cdots+a^{5^{n-1}}.
\]
Alternatively, we could define $\operatorname{tr}(a)$ to be the negation of the coefficient of $x^{n-1}$ in the (characteristic) polynomial
\[
p(x) = (x - a) (x - a^5) (x - a^{25}) \cdots (x - a^{5^{n-1}}).
\]
The
subtrace of $a$ is the coefficient of $x^{n-2}$ in $p(x)$.
The coefficients of $p(x)$ are guaranteed to be elements of $GF(5)$, so the trace and subtrace are elements of $GF(5)$ (i.e., the value is $0,1,\ldots,4$).
|
| (trace,subtrace) |
| $n$
| (0,0)
| (0,1)
(0,4)
| (0,2)
(0,3)
| (1,0)
(2,0)
(3,0)
(4,0)
| (1,1)
(2,4)
(3,4)
(4,1)
| (1,2)
(2,3)
(3,3)
(4,2)
| (1,3)
(2,2)
(3,2)
(4,3)
| (1,4)
(2,1)
(3,1)
(4,4)
|
| 1
| 1
| 0
| 0
| 1
| 0
| 0
| 0
| 0
|
| 2
| 1
| 0
| 2
| 0
| 2
| 2
| 0
| 1
|
| 3
| 1
| 6
| 6
| 6
| 6
| 1
| 6
| 6
|
| 4
| 25
| 30
| 20
| 30
| 25
| 30
| 20
| 20
|
| 5
| 125
| 100
| 150
| 125
| 125
| 125
| 125
| 125
|
| 6
| 625
| 650
| 600
| 625
| 650
| 600
| 600
| 650
|
| 7
| 3025
| 3150
| 3150
| 3150
| 3150
| 3150
| 3150
| 3025
|
Examples
-
Let $GF(5^2)$ be defined by the field extension $GF(5)[x]/(2+x+x^2)$.
The two elements of $GF(5^2)$ with trace 2 and subtrace 4 are $\{4+x, 3+4x\}$.
-
Let $GF(5^2)$ be defined by the field extension $GF(5)[x]/(2+x+x^2)$.
The two elements of $GF(5^2)$ with trace 4 and subtrace 2 are $\{x, 4+4x\}$.
-
Let $GF(5^3)$ be defined by the field extension $GF(5)[x]/(3+2x+3x^2+x^3)$.
The six elements of $GF(5^3)$ with trace 3 and subtrace 1 are $\{2+x+x^2, 3+2x+x^2, 4+3x+2x^2, 3+2x+3x^2, 4+3x+4x^2, 4x+4x^2\}$.
Enumeration (OEIS)
-
Column (0,0) is OEIS A074006.
-
Column (0,1),(0,4) is OEIS A074007.
-
Column (0,2),(0,3) is OEIS A074008.
-
Column (1,0),(2,0),(3,0),(4,0) is OEIS A074009.
-
Column (1,1),(2,4),(3,4),(4,1) is OEIS A074010.
-
Column (1,2),(2,3),(3,3),(4,2) is OEIS A074011.
-
Column (1,3),(2,2),(3,2),(4,3) is OEIS A074012.
-
Column (1,4),(2,1),(3,1),(4,4) is OEIS A074013.