Abstract
By an optimal linear code we mean that it has the highest minimum distance with a prescribed length and dimension. We construct several families of optimal linear codes over the finite field Fp by making use of down-sets generated by one maximal element of Fpn. Moreover, we show that these families of optimal linear codes are minimal and contain relative two-weight linear codes, and have applications to secret sharing schemes and wire-tap channel of type II with the coset coding scheme, respectively.
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