An infinite family of linear codes supporting 4-designs

Abstract

The first linear code supporting a $4$-design was the $[11,6,5]$ ternary Golay code discovered in 1949 by Golay. In the past 71 years, sporadic linear codes holding $4$-designs or $5$-designs were discovered and many infinite families of linear codes supporting $3$-designs were constructed. However, the question as to whether there is an infinite family of linear codes holding an infinite family of $t$-designs for $t\ge 4$ remains open for 71 years. This paper settles this long-standing problem by presenting an infinite family of BCH codes of length $2^{2m+1}+1$ over $\mathrm{GF}(2^{2m+1})$ holding an infinite family of $4$-$(2^{2m+1}+1,6,2^{2m}-4)$ designs. Moreover, an infinite family of linear codes holding the spherical design $S(3,5,4^m+1)$ is presented.

arXiv admin note: substantial text overlap with arXiv:1910.08265

Cited by in corpus (3)

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