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

Abstract

Combinatorial designs are closely related to linear codes. Recently, some near MDS codes were employed to construct <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-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 <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula>-designs for <inline-formula> <tex-math notation="LaTeX">$t \geq 2$ </tex-math></inline-formula>. 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 <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(p^{m})$ </tex-math></inline-formula> holding an infinite family of 3-designs. We then provide an infinite family of almost MDS codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(p^{m})$ </tex-math></inline-formula> holding an infinite family of 2-designs for any field <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(q)$ </tex-math></inline-formula>. In particular, some of these almost MDS codes are near MDS. Second, we present an infinite family of near MDS codes over <inline-formula> <tex-math notation="LaTeX">${\mathrm{ GF}}(2^{m})$ </tex-math></inline-formula> 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.