Infinite Families of 3-Designs and 2-Designs From Almost MDS Codes

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct
$t$
-designs by Ding and Tang, which settles the question as to whether there exists an infinite family of near MDS codes holding an infinite family of
$t$
-designs for
$t \geq 2$
. This paper is devoted to the construction of infinite families of 3-designs and 2-designs from special equations over finite fields. First, we present an infinite family of almost MDS codes over
${\mathrm{ GF}}(p^{m})$
holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over
${\mathrm{ GF}}(p^{m})$
holding an infinite family of 2-designs for any field
${\mathrm{ GF}}(q)$
. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over
${\mathrm{ GF}}(2^{m})$
holding an infinite family of 3-designs by considering the number of roots of a special linearized polynomial. Compared to previous constructions of 3-designs or 2-designs from linear codes, the parameters of some of our designs are new and flexible.