$(r,s)$-Designs in the Nonbinary Johnson Scheme: Constructions and Linear Codes

Using factorizations of complete graphs, we determine the existence conditions for $(2,1)$-designs in the nonbinary Johnson scheme where each $3$-element subset appears exactly once as a support. Furthermore, by utilizing large sets of Steiner triple systems, we construct designs with index $1$ that cannot be obtained from the known construction combining block designs and orthogonal arrays. For linear codes, we express multiplicative character sums over specified nonzero coordinates as sums over the codewords of the dual code. Using this identity along with the rank of a matrix formed by Krawtchouk polynomials, we provide a sufficient condition for the set of all codewords of a given weight to form an $(r,s)$-design. In this matrix, we can utilize not only the rows corresponding to weights absent from the code, but also the rows corresponding to weights for which the set of codewords already forms a design. Moreover, let $q>2$ be a prime power, and let 0 ≠ C Fqⁿ be a linear code with exactly two nonzero weights. For t ≥ 2, the set of all codewords of weight w ≥ t forms a non-empty $(t,t)$-design if and only if the dual minimum distance is at least $t+2$. Consequently, this provides a negative answer to the Wu--Zhou problem concerning two-weight codes with a dual minimum distance of $3$.