Abstract:
Constructs of different classes of finite non-Desargues
translation planes and quasifields closely related. It used by
computer calculations since the middle of last century. We study
semifields of order 32 and quasifields of order 16 of
corresponding translation planes.
It is known that translation planes of any order $p^n$ for a prime p
can be constructed by using a coordinatizing set $W$ of order $n$ over the
field of order $p$. By using a spread set we providing $W$ of structure of quasifield.
The plane is set to be a semifield plane if $W$ is a semifield. The plane is Desargues
if $W$ is a field. It is well-known that semifield planes are isomorphic if and only if their semifields are isotopic.
Structure of quasifields of order $p^n$ has been studied a few, even for small $n$.
In 1960 Kleinfeld classified quasifields of order 16 with kernel of
order 4 and all semifields of order 16 up to isomorphisms. Later
Dempwolf and other completed the classification of all translation planes
of order 16 and 32. We construct 5 semifields of order 32 and 7 quasifields
of order 16 of non-Desargues planes by using their spread sets. For these semifields and for
these quasifields (partially) our main results list for them
introduced orders of all non-zero elements and all subfields.
Keywords:translation planes, spread set, quasifield, semifield, order of element of loop.
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