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

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, 171, 1011, 565

Offset: 1

Author

Joerg Arndt, May 14 2011

Keywords

• nonn
• hard
• more

Comments

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.

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.

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.

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]

Examples

			For n=5 there is just one field polynomial (x^5 + x^4 + x^2 + x + 1),
  for p in {11, 31, 41, 61, 71, 101, 131, ...} (A040160).
For n=7 there is just one field polynomial (x^7 + x^6 + x^4 + x + 1),
  for p in {29, 43, 71, 113, 127, 197,...} (A042967).
For n=11 there are three GNBs:
x^11 + x^10 + x^8 + x^4 + x^3 + x^2 + 1
  for p in {23, 463, 661, 859, 881, 1409, 1453, 2179, ...},
x^11 + x^10 + x^8 + x^5 + x^2 + x + 1
  for p in {67, 89, 353, 727, 947, 1277, 1607, 1783, 1871, ...}, and
x^11 + x^10 + x^8 + x^7 + x^6 + x^5 + 1
  for p in {199, 397, 419, 617, 683, 991, 1123, 2003, 2069, 2113, ...}.
		

Links

• Joerg Arndt, Fxtbook, section 42.9 "Gaussian normal bases", pp. 914-920.

Formula

a(8*n) = 0 (there is no GNB for multiples of eight).