Editing Steiner system (section)

=== Construction from the [[Miracle Octad Generator]] ===
The [[Miracle Octad Generator]] (MOG) is a tool to generate octads, such as those containing specified subsets. It consists of a 4x6 array with certain weights assigned to the rows. In particular, an 8-subset should obey three rules in order to be an octad of S(5,8,24). First, each of the 6 columns should have the same [[Parity (mathematics)|parity]], that is, they should all have an odd number of cells or they should all have an even number of cells. Second, the top row should have the same parity as each of the columns. Third, the rows are respectively multiplied by the weights 0, 1, 2, and 3 over the [[Finite field#Field with four elements|finite field of order 4]], and column sums are calculated for the 6 columns, with multiplication and addition using the [[finite field arithmetic]] definitions. The resulting column sums should form a valid ''[[hexacode]]word'' of the form {{nowrap|(''a'', ''b'', ''c'', ''a'' + ''b'' + ''c'', ''3a'' + ''2b'' + ''c'', ''2a'' + ''3b'' + ''c'')}} where ''a, b, c'' are also from the finite field of order 4. If the column sums' parities don't match the row sum parity, or each other, or if there do not exist ''a, b, c'' such that the column sums form a valid hexacodeword, then that subset of 8 is not an octad of S(5,8,24).

The MOG is based on creating a [[bijection]] (Conwell 1910, "The three-space PG(3,2) and its group") between the 35 ways to partition an 8-set into two different 4-sets, and the 35 lines of the [[Fano plane#Fano three-space|Fano 3-space]] PG(3,2). It is also geometrically related (Cullinane, "Symmetry Invariance in a Diamond Ring", Notices of the AMS, pp A193-194, Feb 1979) to the 35 different ways to partition a 4x4 array into 4 different groups of 4 cells each, such that if the 4x4 array represents a four-dimensional finite [[affine space]], then the groups form a set of parallel subspaces.