A generic construction produces an infinite family of q-ary NMDS codes supporting 2-designs with lengths exceeding q+1 and explicit weight distributions.
Ding, Infinite families of 3-designs from a type of five-weight code, Des
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A Generic Construction of $q$-ary Near-MDS Codes Supporting 2-Designs with Lengths Beyond $q+1$
A generic construction produces an infinite family of q-ary NMDS codes supporting 2-designs with lengths exceeding q+1 and explicit weight distributions.