A143695 - OEIS

%I #11 Mar 06 2013 13:38:33

%S 5,15,35,83,95,495,605,1515,2345,4635,5135,46895,20495,129735,240065,

%T 393179,335405,2125035,1310735,6575675,19010915,15774795,21033005,

%U 220627935,99615005,251842635,614734715,3004955987,1342177295,14604296355,9191328125,25769803707

%N Number of additive cyclic codes over GF(4) of length n.

%D W. C. Huffman, Additive cyclic codes over F_4, Advances in Math. Communication, 2 (2008), 309-343.

%H Joseph Geraci, Frank Van Bussel, <a href="http://arxiv.org/abs/cs/0703129">A theorem on the quantum evaluation of Weight Enumerators for a certain class of Cyclic Codes with a note on Cyclotomic cosets</a>, arXiv:cs/0703129, 2007-2008.

%F Let n=2^z y where y is odd. Let d_0,d_1, ..., d_s be the sizes of the 2-cyclotomic cosets modulo y. Then a(n) = \prod_{i=0}^s \left(1+2^z+\left(\frac{2^{d_i}+1}{2^{d_i}-1}\right)\left(\frac{2^{2^z{d_i}}-1}{2^{d_i}-1}-2^z+2^{2^z{d_i}}-1\right)\right).

%F Also A143696(n) = \prod_{i=0}^s \left(1+\left(\frac{2^{d_i}+1}{2^{d_i}-1}\right)(2^{2^z{d_i}}-1)\right).

%o (PARI)

%o csiz(n, q) = {list = listcreate(n); A = vector(n); for (i=0, n-1, ai = i+1; if (!A[ai], ni = i; nai = ni+1; s = 0; while (! A[nai], A[nai] = 1; s++; ni = lift(Mod(ni*q, n)); nai = ni+1;); listput(list, s););); return (Vec(list));} /* algorithm from arXiv:cs/0703129 */

%o a(n) = {expz = 2^valuation(n, 2); y = n/expz; d = csiz(y, 2); prod(i=1, length(d), 1 + expz + ((2^d[i]+1)/(2^d[i]-1)*((2^(expz*d[i])-1)/(2^d[i]-1) - expz + 2^(expz*d[i])-1)));}

%o \\ _Michel Marcus_, Mar 06 2013

%Y Cf. A143696.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Nov 13 2008, based on email from W. C. Huffman

%E More terms from _Michel Marcus_, Mar 06 2013