A143695 Number of additive cyclic codes over GF(4) of length n.

5, 15, 35, 83, 95, 495, 605, 1515, 2345, 4635, 5135, 46895, 20495, 129735, 240065, 393179, 335405, 2125035, 1310735, 6575675, 19010915, 15774795, 21033005, 220627935, 99615005, 251842635, 614734715, 3004955987, 1342177295, 14604296355, 9191328125, 25769803707

Offset

1,1

References

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

Links

Table of n, a(n) for n=1..32.

Joseph Geraci, Frank Van Bussel, A theorem on the quantum evaluation of Weight Enumerators for a certain class of Cyclic Codes with a note on Cyclotomic cosets, arXiv:cs/0703129, 2007-2008.

Formula

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).

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).

Prog

(PARI)
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 */
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))); }
\\ Michel Marcus, Mar 06 2013

Crossrefs

Cf. A143696.

Sequence in context: A221140 A330911 A322048 * A019531 A321345 A145454

Adjacent sequences: A143692 A143693 A143694 * A143696 A143697 A143698

Keyword

nonn

Author

N. J. A. Sloane, Nov 13 2008, based on email from W. C. Huffman

Extensions

More terms from Michel Marcus, Mar 06 2013

Status

approved