Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 495–507 | Cite as

Types of spreads and duality of the parallelisms of PG(3, 5) with automorphisms of order 13

  • Svetlana TopalovaEmail author
  • Stela Zhelezova
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


A spread is a set of lines of PG(nq) which partition the point set. A parallelism is a partition of the set of all lines by spreads. Empirical data on parallelisms is of interest both from theoretical point of view, and for different applications. Only 51 explicit examples of parallelisms of PG(3, 5) have been known. We construct all (321) parallelisms of PG(3, 5) with automorphisms of order 13 and classify them by the order of their automorphism group, the number of reguli in their spreads and duality. There are no regular ones among them. There are 19 self-dual parallelisms. We also claim that PG(3, 5) has no point-transitive parallelisms.


Spread Parallelism Authomorphism Self-dual parallelisms 

Mathematics Subject Classification

51E23 51E10 05B25 



We are grateful to professor A. Betten from Colorado State University for focusing our attention to invariant 2, and to the anonymous referees whose remarks contributed to a better presentation and motivation of the results.


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Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBASSofiaBulgaria

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