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I am looking for algorithms to construct Perfect Difference Sets.

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.

Given $q$ that satisfies this criteria, we would like to construct a PDS.

  1. Any references to algorithms to build the PDS is appreciated.
  2. If $q$ is prime (or a prime of special form), are there efficient constructions?

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

vvg
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    The $q=m^2+m+1$ criteria indicates a connection with the game called "Dobble" using finite projective geometries with adequate dimensions. – Jean Marie Dec 01 '22 at 10:44
  • I found this essay by A. Chmelnitzki, "Some problems related to Singer sets" (url: http://www.maths.gla.ac.uk/~abartel/docs/AtiyahPrizeEssay.pdf) that describes an approach for generating Singer sets (which are PD sets). – vvg Dec 01 '22 at 13:21

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