Abstract The p -rank of a Steiner quadruple system B is the dimension of the linear span of the set of characteristic vectors of blocks of B over GF ( p ) . We derive a formula for the number of different Steiner quadruple systems of order v and given 2-rank r , r v . Our result extends previous work on enumerating Steiner quadruple systems according to the rank of their codes over GF ( 2 ) , mainly by V.A. Zinoviev and D.V. Zinoviev.

中文翻译：

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非满秩 Steiner 四重系统的个数 S(v,4,3)

摘要 Steiner四元系统B的p秩是B的块在GF(p)上的特征向量集的线性跨度的维数。我们推导出 v 阶不同 Steiner 四重系统的数量并给定 2-rank r , rv 的公式。我们的结果扩展了之前根据其代码在 GF (2) 上的等级枚举 Steiner 四重系统的工作，主要是 VA Zinoviev 和 DV Zinoviev。