Egalitarian Steiner quadruple systems of doubly even order

Abstract

Ordering the blocks of a design, the point sum of an element is the sum of the indices of blocks containing that element. Block labelling for popularity asks for the point sums to be as equal as possible. When all point sums are equal, the system is egalitarian; when point sums differ by at most one, it is almost egalitarian. For Steiner quadruple systems, a doubling construction is adapted to establish that an egalitarian S(3,4,v) exists whenever v≡4,20(mod24) and that an almost egalitarian S(3,4,v) exists whenever v≡8,16(mod24) and v≠8.