number of normal bases

[Max Alekseyev]

Dear Joerg,

I've implemented in PARI/GP the formulae for self-dual (normal) bases
as given in the paper

Dieter Jungnickel, Alfred J. Menezes, Scott A. Vanstone "On the Number
of Self-Dual Bases of GF(q^m) Over GF(q)". Proceedings of the American
Mathematical Society (1990), 109 (1), 23-29.
http://www.math.uwaterloo.ca/~ajmenezes/publications/sdb.pdf

The implementation is available at my PARI/GP scripts page:
http://www.cs.ucsd.edu/users/maxal/gpscripts/

It turns out that the sequence of the number of distinct self-dual
bases of GF(2^n) over GF(2) (i.e., sd(n,2) as denoted in the paper) is
A088437. Here are its first 20 terms:

1, 1, 1, 2, 6, 32, 288, 4608, 130560, 6684672, 621674496,
106099113984, 33421220904960, 19556188689530880, 21359269286705627136,
43743783499173124374528, 168632285389312394463805440,
1227942828363775231508883701760, 16941927202935006869128068433182720,
444122456468619444070070837134825095168

Neil, please extend A088437, provide a reference to the paper above,
and a link to my scripts page for the implementation.

I confirm that sequence of the numbers of distinct self-dual normal
bases of GF(2^n) over GF(2) (i.e., sdn(n,2) as denoted in the paper)
is missing in OEIS. The first 50 terms of this sequence is:

1, 1, 1, 0, 1, 2, 1, 0, 3, 4, 3, 0, 5, 8, 15, 0, 17, 48, 27, 0, 63,
96, 89, 0, 205, 320, 513, 0, 565, 1920, 961, 0, 3267, 4352, 4095, 0,
7085, 13824, 20475, 0, 25625, 64512, 49923, 0, 184275, 182272, 178481,
0, 299593, 839680

Joerg, please confirm that you have got the same numbers and submit
this sequence to OEIS.
Unfortunately, I did not find the definition of the *primitive*
self-dual normal bases in your book (the word "primitive" in the Index
does not even refer to bases). Could you please give a definition of
this term?

Regards,
Max

On Feb 10, 2008 7:22 PM, Joerg Arndt <arndt at jjj.de> wrote:
> I cannot find entries for:
>
> - Number of type-2 optimal normal basis over GF(2^n)
> There is only the entry
> http://www.research.att.com/~njas/sequences/A054639
> "Queneau numbers" (same seq. as requested)
>
>
> - Number of self-dual normal bases over GF(2^n)
> - Number of primitive self-dual normal bases over GF(2^n)
> (see p.883 of http://www.jjj.de/fxt/#fxtbook for these)
>
> Are these really missing?
>
>
>