cp's OEIS Frontend

A190616 Number of normal bases in GF(2^n) that are Gaussian normal bases.

%I A190616 #23 Oct 30 2023 10:38:04
%S A190616 1,1,1,2,1,4,1,0,3,8,3,16,5,16,15,0,17,48,27,128,63,192,89,0,205,637,
%T A190616 171,1011,565
%N A190616 Number of normal bases in GF(2^n) that are Gaussian normal bases.
%C A190616 A type-t Gaussian normal basis (GNB) exists for GF(2^n) if p=n*t+1 is prime and gcd(n,(p-1)/ord(2 mod p))==1.  In practice one finds (for fixed n) infinitely many t corresponding to some GNB.  As there are only finitely many normal bases for fixed n the GNBs for different t are not in general different but correspond to a finite set of field polynomials.  This sequence gives the number of field polynomials (equivalently, mod-2 reduced multiplication matrices) that correspond to some GNB.
%C A190616 The sequence was computed by determining all field polynomials for types t <= n*500 and discarding duplicate polynomials.  Note that there is no guarantee that the used bound (500*n) leads to discovery of all polynomials.
%C A190616 An efficient method to determine (for fixed n) whether two types, say t1 and t2, correspond to the same polynomial would be of great interest.
%C A190616 A computation using the bound t<=2000 gave a(22)=192 (the old value was 191), so the sequence was corrected past that term and truncated after a(29). [_Joerg Arndt_, May 16 2011]
%H A190616 Joerg Arndt, <a href="https://jjj.de/fxt/#fxtbook">Fxtbook</a>, section 42.9 "Gaussian normal bases", pp. 914-920.
%F A190616 a(8*n) = 0 (there is no GNB for multiples of eight).
%e A190616 For n=5 there is just one field polynomial (x^5 + x^4 + x^2 + x + 1),
%e A190616   for p in {11, 31, 41, 61, 71, 101, 131, ...} (A040160).
%e A190616 For n=7 there is just one field polynomial (x^7 + x^6 + x^4 + x + 1),
%e A190616   for p in {29, 43, 71, 113, 127, 197,...} (A042967).
%e A190616 For n=11 there are three GNBs:
%e A190616 x^11 + x^10 + x^8 + x^4 + x^3 + x^2 + 1
%e A190616   for p in {23, 463, 661, 859, 881, 1409, 1453, 2179, ...},
%e A190616 x^11 + x^10 + x^8 + x^5 + x^2 + x + 1
%e A190616   for p in {67, 89, 353, 727, 947, 1277, 1607, 1783, 1871, ...}, and
%e A190616 x^11 + x^10 + x^8 + x^7 + x^6 + x^5 + 1
%e A190616   for p in {199, 397, 419, 617, 683, 991, 1123, 2003, 2069, 2113, ...}.
%K A190616 nonn,hard,more
%O A190616 1,4
%A A190616 _Joerg Arndt_, May 14 2011