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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. 0
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..91.

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).

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].

EXAMPLE

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;

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

CROSSREFS

Column k=0 gives A003473.

Sequence in context: A103525 A294432 A121436 * A088074 A071463 A335191

Adjacent sequences:  A330691 A330692 A330693 * A330695 A330696 A330697

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Dec 25 2019

STATUS

approved

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Last modified January 18 07:55 EST 2022. Contains 350454 sequences. (Running on oeis4.)