Des. Codes Cryptogr. (2018) 86:611–621
A family of semiﬁelds in odd characteristic
· Daniele Bartoli
· Giorgio Faina
· Fernanda Pambianco
Received: 28 July 2016 / Revised: 22 December 2016 / Accepted: 14 February 2017 /
Published online: 1 March 2017
© Springer Science+Business Media New York 2017
Abstract We study a large family of semiﬁelds in odd characteristic, which contains the
commutative Budaghyan–Helleseth semiﬁelds as well as semiﬁelds which are not isotopic
to commutative semiﬁelds. Using a large group of autotopisms we obtain a complete classi-
ﬁcation result in certain parametric subcases.
Keywords Semiﬁelds · Knuth semiﬁelds · Isotopy · Budaghyan–Helleseth semiﬁelds ·
Projective polynomials · Nuclei
Mathematics Subject Classiﬁcation 12K10 · 51E15 · 51A40
A family of presemiﬁelds B( p, m, s, l, t) of order p
where p is an odd prime and m ≥ 3was
studied in . The main ingredient is a Galois automorphism x → x
where σ = p
The remaining parameters are an element l ∈ (F
and a quadruple t =[p
. The family contains the Budaghyan–Helleseth commutative semiﬁelds.
The special case of order 729 = 3
is studied in .
In the present paper we introduce a new type of isotopism (linear isotopy, Proposition 3)
relating those presemiﬁelds. This leads to a group GL(2, p
(the restricted isotopy
group of Sect. 5) acting on the quadruples t such that quadruples in the same orbit yield
isotopic presemiﬁelds B( p, m, s, l, t ) (given p, m, s, l). We then specialize to the ternary
case and prove that in each of the cases p = 3, s = 1, m ≥ 3and p = 3, s = 2, m ≥ 3
Communicated by M. Lavrauw.
Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931,
Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, 06123 Perugia, Italy