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A135488 Number of distinct self-dual normal bases for GF(2^n) over GF(2) 2
1, 1, 1, 0, 1, 2, 1, 0, 3, 4, 3, 0, 5, 8, 15, 0, 17, 48, 27, 0, 63, 96, 89, 0, 205, 320, 513, 0, 565, 1920, 961, 0, 3267, 4352, 4095, 0, 7085, 13824, 20475, 0, 25625, 64512, 49923, 0, 184275, 182272, 178481, 0, 299593, 839680, 1105425, 0, 1266205, 4202496, 3145725, 0, 7105563, 9256960, 9099507, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

LINKS

Table of n, a(n) for n=1..60.

Max Alekseyev, PARI scripts

Joerg Arndt, Matters Computational (The Fxtbook), see p. 910.

Dieter Jungnickel, Alfred J. Menezes and Scott A. Vanstone, On the Number of Self-Dual Bases of GF(q^m) Over GF(q), Proc. Amer. Math. Soc. 109 (1990), 23-29.

PROG

(PARI)

/* based on http://home.gwu.edu/~maxal/gpscripts/nsdb.gp by Max Alekseyev */

sdn(m, p) =

/* Number of distinct self-dual normal bases of GF(p^m) over GF(p) where p is prime */

{

    local(F, f, g, s, c, d);

    if ( p==2 && m%4==0, return(0) );

    if ( !(m%p), /* p divides m */

        s = m\p;

        return( p^((p-1)*(s+(s*(p+1))%2)/2-1) * sdn(s, p) );

    , /* else */

        F = factormod( (x^m - 1)/(x - 1), p );

        c = d = [];

        for (i=1, matsize(F)[1],

            f = lift(F[i, 1]);

            g = polrecip(f);

            if ( f==g,  c = concat( c, vector(F[i, 2], j, poldegree(f)/2) ); );

            if ( lex(Vec(f), Vec(g))==1 ,

                d = concat( d, vector(F[i, 2], j, poldegree(f)) );

            );

        );

        return( 2^(p%2) * prod(i=1, #c, p^c[i] + 1) * prod(j=1, #d, p^d[j] - 1) / m );

    );

}

vector(66, n, sdn(n, 2)) /* Joerg Arndt, Jul 03 2011 */

CROSSREFS

Cf. A088437.

Sequence in context: A094301 A221542 A221463 * A099493 A088523 A222211

Adjacent sequences:  A135485 A135486 A135487 * A135489 A135490 A135491

KEYWORD

nonn

AUTHOR

Max Alekseyev, Feb 11 2008

STATUS

approved

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Last modified September 26 10:59 EDT 2017. Contains 292518 sequences.