  A finite subset $Y$ on the unit sphere $S^{n-1} \subseteq \mathbb{R}^n$ is
called a spherical design of harmonic index $t$, if the following condition is
satisfied: $\sum_{\mathbf{x}\in Y}f(\mathbf{x})=0$ for all real homogeneous
harmonic polynomials $f(x_1,\ldots,x_n)$ of degree $t$. Also, for a subset $T$
of $\mathbb{N} = \{1,2,\cdots \}$, a finite subset $Y\subset S^{n-1}$ is called
a spherical design of harmonic index $T,$ if $\sum_{\mathbf{x}\in
Y}f(\mathbf{x})=0$ is satisfied for all real homogeneous harmonic polynomials
$f(x_1,\ldots,x_n)$ of degree $k$ with $k\in T$.
  In the present paper we first study Fisher type lower bounds for the sizes of
spherical designs of harmonic index $t$ (or for harmonic index $T$). We also
study 'tight' spherical designs of harmonic index $t$ or index $T$. Here
'tight' means that the size of $Y$ attains the lower bound for this Fisher type
inequality. The classification problem of tight spherical designs of harmonic
index $t$ was started by Bannai-Okuda-Tagami (2015), and the case $t = 4$ was
completed by Okuda-Yu (2015+). In this paper we show the classification
(non-existence) of tight spherical designs of harmonic index 6 and 8, as well
as the asymptotic non-existence of tight spherical designs of harmonic index
$2e$ for general $e\geq 3$. We also study the existence problem for tight
spherical designs of harmonic index $T$ for some $T$, in particular, including
index $T = \{8,4\}$. We use (i) the linear programming method by Delsarte, (ii)
the detailed information on the locations of the zeros as well as the local
minimum values of Gegenbauer polynomials, (iii) the generalization by Hiroshi
Nozaki of the Larman-Rogers-Seidel theorem on $2$-distance sets to $s$-distance
sets, (iv) the theory of elliptic diophantine equations, and (v) the
semidefinite programming method of eliminating some $2$-angular line systems
for small dimensions.
