I have been reading on Perfect Difference Sets and it has been fascinating. I have an open MSE question here seeking references for algorithms to construct PD sets. This is a different question seeking review of an algorithm proposal and proving related claims.
Definition 1. Perfect Difference Sets (PDS) of order $m+1$ are a set of residues $\{d_1,d_2,\cdots,d_{m+1}\} \pmod{q}$ such that every non-zero residue modulo $q$ can be uniquely represented by $d_i−d_j \pmod{q}$ where $d_i,d_j$ are members of the PDS. Singer [Singer1938] gave the criteria that it is necessary that $q=m^2+m+1$ and sufficient that $m=p^g$, a prime power in order for a PDS to exist.
These are also known as PDS of Singer type (other PDS families seek $\lambda$ different representations for non-zero residues).
Setup. Let $A = \{a_0, a_1, a_2, \cdots, a_{q-1}\}$ be a set of residues modulo $q$ with $q = m^2 + m + 1, m = p^g$, a prime power. Let $A_+ = \{x: x \text{ is a non-zero residue modulo $q$} \}$. We have $A_+ \subset A$. If $\Delta$ is a PDS modulo $q$ then the set of $m$ congruences given by
$$ \begin{align} d_i - d_j \equiv a_k \text{ where $1 \le i,j \le m+1, a_k \in A_+$} \end{align} $$
must have a solution. The problem of constructing the PDS then becomes one of choosing $m$ different elements $a_k \in A_+$ so that we have a solution. This gives us $m$ congruences in $m+1$ variables $d_1, d_2, \cdots, d_{m+1}$. This system is still overdefined. However, we are free to choose one of the $d_i$ to be any value modulo $q$ to obtain a system of $m$ congruence relations in $m$ variables. Without loss of generality, take $d_0 = 0$.
One of the constraints of a PDS of Singer type is that the non-zero residues have a unique representation. Therefore, if we consider any three congruences from the system of congruences involving any four variables $d_u, d_v, d_w, d_x$, we have
$$\begin{align} d_u - d_v \equiv a_{r} \pmod{q}\\ d_v - d_w \equiv a_{s} \pmod{q}\\ d_w - d_x \equiv a_{t} \pmod{q} \end{align}$$
Suppose $a_{r}, a_{s}, a_{t}$ form an Arithmetic Progression modulo $q$ with common difference $\delta$. Then we get $d_u + d_x = a_{r} - a_{s} - a_{t}$. But by definition of the PDS, $d_u - d_x = a_{\gamma}$ for some $a_{\gamma} \in A_+$. So, the system is no longer linearly independent.
Claim 1. A necessary condition for the PDS to exist is that no three elements $a_{r}, a_{s}, a_{t} \in A_+$ occuring in the RHS of the system of congruences may form an arithmetic progression.
Proof. WLOG, if three terms in the RHS $a_{r} < a_{s} < a_{t}$ form an AP then $$a_{s} - a_{r} = a_{t} - a_{s} = \delta.$$ This means the non-zero common difference $\delta$ has two representations which contradicts the unique representation condition for non-zero residues mod $q$ as difference of elements from the PDS.
QED
Claim 2. This condition is also sufficient to ensure the solution $d_1, d_2, \cdots, d_{m+1}$ is a PDS.
Question: Are both Claim 1 and Is Claim 2 true?
Context: As mentioned in the opening paragaph, I am seeking algorithm references for constructing PDS in the linked question. The claims in this question, if true, would help with building an algorithm to construct a PDS. Such an algorithm would make $m+1$ choices of $d_i \in A$ and each choice $\alpha$ should satisfy the condition in Claim 1 of no two previously selected elements form an AP along with the chosen $\alpha$.
References
[Singer1938]: J. Singer, "A Theorem in Finite Projective Geometry and Some Applications to Number Theory," Transactions of the American Mathematical Society, vol. 43, no. 2, p. pp. 377–85, 1938. URL (accessed Dec 1, 2022): https://www.ams.org/journals/tran/1938-043-03/S0002-9947-1938-1501951-4/S0002-9947-1938-1501951-4.pdf
