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… Click to show full abstract
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.
