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%I #31 Dec 26 2019 17:21:50
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%C 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).
%H Zülfükar Saygi, Ernist Tilenbaev, Çetin Ürtiş, <a href="https://doi.org/10.3906/mat-1805-113">On the number of k-normal elements over finite fields</a>, Turk J Math., (2019) 43.795.812.
%H David Thompson, <a href="https://pdfs.semanticscholar.org/d0b2/ce1a4b89d3198cc0444e54c238d459a16a22.pdf">Something about normal bases over finite fields</a>, Existence and properties of k-normal elements over finite fields, Slides, (2013).
%F 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].
%e Triangle begins
%e 1;
%e 2, 1;
%e 3, 3, 1;
%e 8, 4, 2, 1;
%e 15, 15, 0, 0, 1;
%e 24, 12, 18, 3, 5, 1;
%e 49, 49, 0, 14, 14, 0, 1;
%e 128, 64, 32, 16, 8, 4, 2, 1;
%e 189, 189, 63, 63, 0, 0, 3, 3, 1;
%e 480, 240, 240, 0, 30, 15, 15, 0, 2, 1;
%e 1023, 1023, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 1536, 768, 768, 384, 384, 96, 96, 24, 26, 7, 5, 1;
%e 4095, 4095, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%t Needs["FiniteFields`"];Table[b = Map[GF[2^n][#] &, Tuples[{0, 1}, n]];
%t Table[Count[Table[MatrixRank[Table[b[[i]]^(2^k), {k, 0, n - 1}][[All, 1]],
%t Modulus -> 2], {i, 2, 2^n}], k], {k, 1, n}] // Reverse, {n, 1, 8}] // Grid
%Y Column k=0 gives A003473.
%K nonn,tabl
%O 1,2
%A _Geoffrey Critzer_, Dec 25 2019
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