The links below give the kernel $k[1..7]$ of $h_1$ over $\mathbb{Z}_8$.
There are $2^{18} = 262,144$ kernel elements in total.
A typical line in this table looks like 5 2 3 0 3 1 7,
meaning that the polynomial
$(1+z)^5 (1+2z)^2 (1+3z)^3 (1+4z)^0 (1+5z)^3 (1+6z)^1 (1+7z)^7$
(mod 8) is of the form $1 + O(z^2)$.
The exact value is:
$1+z^2+2z^3+4z^6+6z^8+6z^{10}+6z^{11}+4z^{12}+4z^{14}+5z^{16}+z^{18}+2z^{19}+4z^{20}$.
Links: Z8n1kernel.txt or zipped
Z8n1kernel.zip.
The links below give the kernel $k[1..7]$ of $h_2$ over $\mathbb{Z}_8$.
There are $2^{20} = 1,048,576$ kernel elements in total.
A typical line in this table looks like 15 4 13 6 5 12 15,
meaning that the polynomial
$(1+z)^{15} (1+2z)^4 (1+3z)^{13} (1+4z)^6 (1+5z)^5 (1+6z)^{12} (1+7z)^{15}$
(mod 8) is of the form $1 + O(z^4)$.
The corresponding expansion up to $O(z^{32})$ is:
$1+4z^4+2z^8+4z^{12}+7z^{16}+4z^{24}+O(z^{32})$.
Links: Z8n2kernel.txt or zipped
Z8n2kernel.zip.
Here there are $2^{24} = 16,777,216$ kernel elements in total.
In this case the kernel is the interleaved product of $O_3$ and $E_3$.
For example, adding the kernel element 24 0 16 0 0 24 from $O_3$ and
0 3 0 1 0 1 0 from $E_3$ gives 24 3 16 1 0 1 24,
meaning that the polynomial $(1+z)^{24} (1+2z)^3 (1+3z)^{16} (1+4z) (1+6z) (1+7z)^{24} \bmod{8}$
is of the form $1 + O(z^8)$.
Links: See $n=3$ below.