The Combinatorial Object Server

Elements of $GF(3^n)$ with given trace and subtrace

If $a$ is an element of $GF(3^n)$, then the (absolute) trace of $a$ is \[ \operatorname{tr}(a)=a+a^3+a^9+\cdots+a^{3^{n-1}}. \] Alternatively, we could define $\operatorname{tr}(a)$ to be the negation of the coefficient of $x^{n-1}$ in the (characteristic) polynomial \[ p(x) = (x - a) (x - a^3) (x - a^9) \cdots (x - a^{3^{n-1}}). \] 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 $GF(3)$, so the trace and subtrace are elements of $GF(3)$ (i.e., the value is 0, 1 or 2).

	(trace,subtrace)
$n$	(0,0)	(0,1)	(0,2)	(1,0) (2,0)	(1,1) (2,1)	(1,2) (2,2)
1	1	0	0	1	0	0
2	1	2	0	0	1	2
3	3	0	6	3	3	3
4	9	12	6	9	12	6
5	21	30	30	30	21	30
6	99	72	72	81	81	81
7	225	252	252	225	252	252
8	729	702	756	756	729	702
9	2187	2268	2106	2187	2187	2287
10	6561	6480	6642	6561	6480	6642
11	19845	19602	19602	19602	19845	19602
12	58563	59292	59292	59049	59049	59049
13	177633	176904	176904	177633	176904	176904
14	531441	532170	530712	530712	531441	532170
15	1594323	1592136	1596510	1594323	1594323	1594323
16	4782969	4785156	4780782	4782969	4785156	4780782

Examples

• Let $GF(3^2)$ be defined by the field extension $GF(3)[x]/(2+x+x^2)$. The two elements of $GF(3^2)$ with trace 0 and subtrace 1 are $\{2+x, 1+2x\}$.
• Let $GF(3^2)$ be defined by the field extension $GF(3)[x]/(2+x+x^2)$. The two elements of $GF(3^2)$ with trace 1 and subtrace 2 are $\{1+x, 2x\}$.
• Let $GF(3^3)$ be defined by the field extension $GF(3)[x]/(1+x+2x^2+x^3)$. The three elements of $GF(3^3)$ with trace 2 and subtrace 1 are $\{2x, 1+x^2, 1+x+2x^2\}$.

Enumeration (OEIS)

• Column (0,0) is OEIS A074000.
• Column (0,1) is OEIS A074001.
• Column (0,2) is OEIS A074002.
• Column (1,0),(2,0) is OEIS A074003.
• Column (1,1),(2,1) is OEIS A074004.
• Column (1,2),(2,2) is OEIS A074005.