number of normal bases

[Joerg Arndt]

Hi,

* Max Alekseyev <maxale at gmail.com> [Feb 13. 2008 10:43]:
> Dear Joerg,
> 
> I've implemented in PARI/GP the formulae for self-dual (normal) bases
> as given in the paper

Thanks for that!

> 
> [...]
> 
> 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

Seq. agrees with mine as far as computed (exhaustive search).
(top of fig.40.6-I on p.886)

> and submit this sequence to OEIS.

I will.

> 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?

Basis primitive <==> field polynomial (minimal polynomial of any of
the base elements) primitive.
The numbers are given at bottom of fig.40.6-I on p.886.
I'll add that to the text (and index: prim.element with finite field).

Thanks for the detailed answer!

P.S.: I'll also add the seq. of values n such that a typ-2 optimal
normal basis exists ("number of" was nonesense).