Elements of $GF(2^n)$ with given trace and subtrace

If $a$ is an element of $GF(2^n)$, then the (absolute) trace of $a$ is \[ \operatorname{tr}(a)=a+a^2+a^4+\cdots+a^{2^{n-1}}. \] Alternatively, we could define $\operatorname{tr}(a)$ to be the negation of the coefficient of $x^{n-1}$ in the (characteristic) polynomial \[ p(x) = (x - a) (x - a^2) (x - a^4) \cdots (x - a^{2^{n-1}}). \] 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 $GF(2)$, so the trace and subtrace are elements of $GF(2)$ (i.e., the value is 0 or 1).

binomial coeffient sum (trace,subtrace)
n 0 1 2 3 (0,0) (0,1) (1,0) (1,1)
0 1 0 0 0 0 1 0 0
1 1 1 0 0 1 0 1 0
2 1 2 1 0 1 1 0 2
3 1 3 3 1 1 3 3 1
4 2 4 6 4 6 2 4 4
5 6 6 10 10 6 10 6 10
6 16 12 16 20 16 16 20 12
7 36 28 28 36 36 28 28 36
8 72 64 56 64 56 72 64 64
9 136 136 120 120 136 120 136 120
10 256 272 256 240 256 256 240 272
11 496 528 528 496 496 528 528 496
12 992 1024 1056 1024 1056 992 1024 1024
13 2016 2016 2080 2080 2016 2080 2016 2080
14 4096 4032 4096 4160 4096 4096 4160 4032
15 8256 8128 8128 8256 8256 8128 8128 8256
16 16512 16384 16256 16384 16256 16512 16384 16384
17 32896 32896 32640 32640 32896 32640 32896 32640
18 65536 65792 65536 65280 65536 65536 65280 65792
19 130816 131328 131328 130816 130816 131328 131328 130816
20 261632 262144 262656 262144 262656 261632 262144 262144
21 523776 523776 524800 524800 523776 524800 523776 524800
22 1048576 1047552 1048576 1049600 1048576 1048576 1049600 1047552
23 2098176 2096128 2096128 2098176 2098176 2096128 2096128 2098176
24 4196352 4194304 4192256 4194304 4192256 4196352 4194304 4194304

	binomial coeffient sum		(trace,subtrace)
n	0	1	2	3	(0,0)	(0,1)	(1,0)	(1,1)
0	1	0	0	0	0	1	0	0
1	1	1	0	0	1	0	1	0
2	1	2	1	0	1	1	0	2
3	1	3	3	1	1	3	3	1
4	2	4	6	4	6	2	4	4
5	6	6	10	10	6	10	6	10
6	16	12	16	20	16	16	20	12
7	36	28	28	36	36	28	28	36
8	72	64	56	64	56	72	64	64
9	136	136	120	120	136	120	136	120
10	256	272	256	240	256	256	240	272
11	496	528	528	496	496	528	528	496
12	992	1024	1056	1024	1056	992	1024	1024
13	2016	2016	2080	2080	2016	2080	2016	2080
14	4096	4032	4096	4160	4096	4096	4160	4032
15	8256	8128	8128	8256	8256	8128	8128	8256
16	16512	16384	16256	16384	16256	16512	16384	16384
17	32896	32896	32640	32640	32896	32640	32896	32640
18	65536	65792	65536	65280	65536	65536	65280	65792
19	130816	131328	131328	130816	130816	131328	131328	130816
20	261632	262144	262656	262144	262656	261632	262144	262144
21	523776	523776	524800	524800	523776	524800	523776	524800
22	1048576	1047552	1048576	1049600	1048576	1048576	1049600	1047552
23	2098176	2096128	2096128	2098176	2098176	2096128	2096128	2098176
24	4196352	4194304	4192256	4194304	4192256	4196352	4194304	4194304

Examples

Let $GF(2^2)$ be defined by the field extension $GF(2)[x]/(1+x+x^2)$. The two elements of $GF(2^2)$ with trace 1 and subtrace 1 are $\{x, 1+x\}$.
Let $GF(2^3)$ be defined by the field extension $GF(2)[x]/(1+x^2+x^3)$. The three elements of $GF(2^3)$ with trace 1 and subtrace 0 are $\{x, x^2, 1+x+x^2\}$.
Let $GF(2^4)$ be defined by the field extension $GF(2)[x]/(1+x+x^4)$. The two elements of $GF(2^4)$ with trace 0 and subtrace 1 are $\{x+x^2,1+x+x^2\}$.

Enumeration (OEIS)

The column labelled 0 has entries $\binom{n}{0}+\binom{n}{4}+\binom{n}{8}+\cdots$, which is also equal to the (0,1) column when $n$ is even and the (0,0) column when $n$ is odd. Column 0 is OEIS A038503.
The column labelled 1 has entries $\binom{n}{1}+\binom{n}{5}+\binom{n}{9}\cdots$, which is also equal to the (1,1) column when $n$ is even and the (1,0) column when $n$ is odd. Column 1 is OEIS A038504.
The column labelled 2 has entries $\binom{n}{2}+\binom{n}{6}+\binom{n}{10}+\cdots$, which is also equal to the (0,0) column when $n$ is even and the (0,1) column when $n$ is odd. Column 2 is OEIS A038505.
The column labelled 3 has entries $\binom{n}{3}+\binom{n}{7}+\binom{n}{11}+\cdots$, which is also equal to the (1,0) column when $n$ is even and the (1,1) column when $n$ is odd. Column 3 is OEIS A038506.
Column (0,0) is OEIS A038518.
Column (0,1) is OEIS A038519.
Column (1,0) is OEIS A038520.
Column (1,1) is OEIS A038521.
For proofs of these facts, see [CMR+03].
The entries in the above table can be expressed as the sum of two powers of 2. Let $m=\lfloor n/2\rfloor$. Then the entry of the first four columns are $2^{n-2}+s2^{m-1}$, where $s$ is 0, +1, or -1, depending on the corresponding entry from the table below.
coeff. sum
$n$ mod 8 0 1 2 3
0 + 0 - 0
1 + + - -
2 0 + 0 -
3 - + + -
4 - 0 + 0
5 - - + +
6 0 - 0 +
7 + - - +

	coeff. sum
$n$ mod 8	0	1	2	3
0	+	0	-	0
1	+	+	-	-
2	0	+	0	-
3	-	+	+	-
4	-	0	+	0
5	-	-	+	+
6	0	-	0	+
7	+	-	-	+

References

[CMR+03] K. Cattell, C. R. Miers, F. Ruskey, J. Sawada, and M. Serra. The number of irreducible polynomials over GF(2) with given trace and subtrace. J. Combin. Math. Combin. Comput., 47:31–64, 2003.