  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 $t\geq 4$ 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.
