Block of a Steiner system

Solution

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 11 / 1 / i / Solution

Put $m=\binom{k}{2}$, take odd $q\equiv1\pmod{2m}$, and let $H$ be the index-$m$ multiplicative <subgroup>. If a $k$-set $A$ has its unordered pair differences in distinct $H$-cosets, choose $S\subset H$ containing exactly one element from each pair $\{h,-h\}$. Then $\{sA:s\in S\}$ is a <difference family for a Steiner 2-design>. Indeed one unordered pair supplies each <coset>, and its two difference orientations multiplied by $S$ cover that <coset> once. Translations yield a $2$-$(q,k,1)$ <Steiner system>. The <cyclotomic pattern extension by a second-moment bound> supplies suitable $A$ for all sufficiently large $q$ in this progression.

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 Incoming links: Cyclotomic construction of a Steiner 2-design