1 research outputs found
Efficient Methods in Counting Generalized Necklaces
It is shown in [7] by Venkaiah in 2015 that a category of the number of
generalized can be computed using the expression \begin{equation*} e(n, q) =
\frac{1}{(q-1) ord(\lambda) n} \sum^{ord(\lambda)n}_{\substack{t \in
\mathbb{F}_q \setminus \{0\}, i=1 \\ t^{\frac{n}{\gcd(n, i)}}
\lambda^{\frac{i}{\gcd(n,i)}} = 1}}(q^{\gcd(n,i)} - 1) + 1 \end{equation*}
where (number of colors) is the size of the prime field ,
is the constant of the consta-cyclic shift, is the length of the
necklace. However, direct evaluation of this expression requires, apart from
the computations, exponentiations and
multiplications, at most
exponentiations and at most additions and hence
computationally intensive. This note discusses various other ways of evaluating
the expression and tries to throw some light on amortizing the amount of
computation.Comment: 8 page