A330694 Triangular array read by rows. T(n,k) is the number of k-normal elements in GF(2^n), n >= 1, 0 <= k <= n-1.

1, 2, 1, 3, 3, 1, 8, 4, 2, 1, 15, 15, 0, 0, 1, 24, 12, 18, 3, 5, 1, 49, 49, 0, 14, 14, 0, 1, 128, 64, 32, 16, 8, 4, 2, 1, 189, 189, 63, 63, 0, 0, 3, 3, 1, 480, 240, 240, 0, 30, 15, 15, 0, 2, 1, 1023, 1023, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1536, 768, 768, 384, 384, 96, 96, 24, 26, 7, 5, 1, 4095, 4095, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Author

Geoffrey Critzer, Dec 25 2019

Keywords

• nonn
• tabl

Comments

Let g(x) = Sum_{i=0..n-1} g_i x^i in GF(q)[x]. Define an action on the algebraic closure of GF(q) by g*a = Sum_{i=0..n-1} g_i a^(q^i). The annihilator of a is an ideal and is generated by a polynomial g of minimum degree. If deg(g) = n-k then a is k-normal.

An element a in GF(q^n) is 0-normal if {a, a^q, a^(q^2), ..., a^(q^(n-1))} is a basis for GF(q^n) over GF(q).

Examples

			Triangle begins
     1;
     2,    1;
     3,    3,   1;
     8,    4,   2,   1;
    15,   15,   0,   0,   1;
    24,   12,  18,   3,   5,  1;
    49,   49,   0,  14,  14,  0,  1;
   128,   64,  32,  16,   8,  4,  2,  1;
   189,  189,  63,  63,   0,  0,  3,  3,  1;
   480,  240, 240,   0,  30, 15, 15,  0,  2, 1;
  1023, 1023,   0,   0,   0,  0,  0,  0,  0, 0, 1;
  1536,  768, 768, 384, 384, 96, 96, 24, 26, 7, 5, 1;
  4095, 4095,   0,   0,   0,  0,  0,  0,  0, 0, 0, 0, 1;
		

Links

• S. Huczynska, G. Mullen, D. Panario, and D. Thomson, Existences and properties of k-normal elements over finite fields, Finite Fields and Their Applications, 24 (2013), 170-183.
• Zülfükar Saygi, Ernist Tilenbaev, Çetin Ürtiş, On the number of k-normal elements over finite fields, Turk J Math., (2019) 43.795.812.
• David Thompson, Something about normal bases over finite fields, Existence and properties of k-normal elements over finite fields, Slides, (2013).

Crossrefs

Column k=0 gives A003473.

Programs

Mathematica

Needs["FiniteFields`"];Table[b = Map[GF[2^n][#] &, Tuples[{0, 1}, n]];
  Table[Count[Table[MatrixRank[Table[b[[i]]^(2^k), {k, 0, n - 1}][[All, 1]],
       Modulus -> 2], {i, 2, 2^n}], k], {k, 1, n}] // Reverse, {n, 1, 8}] // Grid

Formula

T(n, k) = Sum_{h | x^n-1, deg(h) = n-k} phi_2(h) where phi_2(h) is the generalized Euler phi function and the polynomial division is in GF(2)[x].