Suppose
$$ P \subseteq \{1,2,\dots,N\},\quad |P| = K $$
We calculate the differences as:
$$ d=p_i-p_j\mod N,\quad i\ne j $$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 0, 1, 2, \dots , N − 1$), then we have a set
$$ D=\{a_0,a_1,a_2,...,a_{N-1} \} $$
Let's define a valid $D$ as a $D$ which is resulted from a points set $P$ (a $D$ which there exists some $P$ leading to it).
Reconstructing $P$ from $D$ is called beltway reconstruction and there are some algorithms for it, but I'm looking for some criteria that indicate the given $D$ is not a valid one.
One is $\sum_{d=1}^{N-1} a_d = K(K-1)$, i.e. if this equality doesn't hold for the given $D$ that $D$ is not a valid one.
The other is $a_i = a_{N-i}$ for $i=1,2,\cdots,N-1$
EDIT: consider these two sets of $D$'s ($a_0=K$ is included, too)
$(N,K) = (10,5)$:
$$\color{green}{\{5, 3, 2, 2, 2, 2, 2, 2, 2, 3\}}$$
$$\{5,2, 3, 2, 2, 2, 2, 2, 3, 2\}$$
$$\color{green}{\{5, 2, 2, 3, 2, 2, 2, 3, 2, 2\}}$$
$$\{5,2, 2, 2, 3, 2, 3, 2, 2, 2\}$$
$(N,K) = (11,8)$ :
$$\color{green}{ \{ 8, 6 , 6, 6 , 5, 5 , 5, 5 , 6, 6 , 6 \}}$$
$$\{ 8 , 6 , 6 , 5 , 6 , 5 , 5 , 6 , 5 , 6 , 6\}$$
$$\{ 8 , 6 , 6 , 5 , 5 , 6 , 6 , 5 , 5 , 6 , 6\}$$
$$\color{green}{\{ 8 , 6, 5 , 6, 6 , 5, 5 , 6, 6 , 5, 6 \}}$$
$$\{ 8, 6 ,5, 6 ,5, 6 ,6, 5 ,6, 5 ,6\}$$
$$\color{green}{\{ 8 , 6 , 5 , 5 , 6 , 6 , 6 , 6 , 5 , 5 , 6 \}}$$
$$\{ 8 , 5 , 6 , 6 , 6 , 5 , 5 , 6 , 6 , 6 , 5\}$$
$$\color{green}{\{ 8 , 5 , 6 , 6 , 5 , 6 , 6 , 5 , 6 , 6 , 5 \}}$$
$$\color{green}{\{ 8 , 5 , 6 , 5 , 6 , 6 , 6 , 6 , 5 , 6 , 5 \}}$$
$$\{ 8 ,5, 5 ,6, 6 ,6, 6 ,6, 6 ,5, 5\}$$
Only green ones are permissible and the others are not, while they satisfy symmetric criteria suggested thanks to Thomas. So there must be some asymmetric criteria.
Are there any other criteria?
This question is related, if helps.