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2020 | OriginalPaper | Chapter

Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality

Authors : Eduard Vatutin, Alexey Belyshev

Published in: Supercomputing

Publisher: Springer International Publishing

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Abstract

The article describes computational experiments aimed to enumerating the number of orthogonal diagonal Latin squares for general and special types of orthogonality. General type of orthogonality can be verified using Euler-Parker method, corresponding the number of main classes of orthogonal diagonal Latin squares, the number of normalized orthogonal diagonal Latin squares and total number of orthogonal diagonal Latin squares of general type form previously unknown numerical series A330391, A305570 and A305571 (calculated up to order 8) has been added to OEIS. Self-orthogonal (SODLS), doubly self-orthogonal (DSODLS) and extended self-orthogonal diagonal Latin squares (ESODLS) form the set of special types of orthogonality. For each of these types corresponding numerical series was calculated and published in OEIS with numbers A329685, A287761, A287762 (SODLS, up to order 10), A333366, A333367, A333671 (DSODLS, up to order 10) and A309210, A309598, A309599 (ESODLS, up to order 8). Values for orders 1–8 were obtained by analyzing the complete lists of canonical forms of the main classes of orthogonal DLSs obtained by the authors by exhaustive search. Values for order 9 were derived from the SODLS list of order 9 provided by Harry White. Values for order 10 were obtained by analyzing the list of SOLS of order 10, available online (van Vuuren et al.). The values obtained confirm the similar values for SODLS and DSODLS obtained previously by Francis Gaspalou and partially published by Harry White. For some of the obtained numerical values previously unknown mathematical relations are empirically established.

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Title: Enumerating the Orthogonal Diagonal Latin Squares of Small Order for Different Types of Orthogonality
Authors: Eduard Vatutin
Alexey Belyshev
Publisher: Springer International Publishing
Book: Supercomputing
Print ISBN: 978-3-030-64615-8

Electronic ISBN: 978-3-030-64616-5

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-64616-5_50

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