A note on MDS codes, n-arcs and complete designs
A generalized incidence matrix of a design over GF(q) is any matrix obtained from the (0, 1)-incidence matrix by replacing ones with nonzero elements from GF(q). The dimension d(q) of a design D over GF(q) is defined as the minimum value of the q-rank of a generalized incidence matrix of D. It is proved that the dimension d(q) of the complete design on n points having as blocks all w-subsets, is greater that or equal to n- w + 1, and the equality d(q) = n- w + 1 holds if and only if there exists an [n, n- w + 1, w] MDS code over GF(q), or equivalently, an n-arc in PG(w - 2, q).
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