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Designs, Codes and Cryptography

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01.04.2015

On point-transitive and transitive deficiency one parallelisms of PG \((3,4)\)

verfasst von: Svetlana Topalova, Stela Zhelezova

Erschienen in: Designs, Codes and Cryptography | Ausgabe 1/2015

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Abstract

A parallelism in PG \((n,q)\) is point-transitive if it has an automorphism group which is transitive on the points. If the automorphism group fixes one spread and is transitive on the remaining spreads, the parallelism corresponds to a transitive deficiency one parallelism. It is known that there are three types of spreads in PG \((3,4)\)—regular, subregular and aregular. A parallelism is regular if all its spreads are regular. In PG \((3,4)\) no point-transitive parallelisms, no regular ones, and no transitive deficiency one parallelisms have been known. Both point-transitive parallelisms and transitive deficiency one parallelisms must have automorphisms of order 5. We construct all 32,048 nonisomorphic parallelisms with automorphisms of order 5 and classify them by the orders of their automorphism groups and by the types of their spreads. There are 31,832 parallelisms with an automorphism group fixing exactly one spread. Only for four of them the automorphism group is transitive on the remaining spreads. Among the parallelisms we construct there are no regular ones. There are 4,124 parallelisms with automorphisms of order 5 without fixed points, but none of them is point-transitive.

Vorheriger Artikel Comment on “Identity-based non-interactive key distribution with forward security”

Nächster Artikel Xing–Ling codes, duals of their subcodes, and good asymmetric quantum codes

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Titel: On point-transitive and transitive deficiency one parallelisms of PG
verfasst von: Svetlana Topalova
Stela Zhelezova
Publikationsdatum: 01.04.2015
Verlag: Springer US
Erschienen in: Designs, Codes and Cryptography / Ausgabe 1/2015
Print ISSN: 0925-1022
Elektronische ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-013-9887-3

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On point-transitive and transitive deficiency one parallelisms of PG \((3,4)\)

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