A *t*-(*v*,*k*,λ)-design D=(P,B) is a pair consisting of a set P of **points** and a collection B of *k*-element subsets of P, called **blocks**, such that the number *r* of blocks that contain any point *p* in *P* is independent of *p*, and the number λ of blocks that contain any given *t*-element subset *T* of P is independent of the choice of *T*. The numbers *v* (the number of elements of *P*), *b* (the number of blocks), *k*, *r*, λ, and *t* are the **parameters** of the design. The parameters must satisfy several combinatorial identities, for example:

where is the number of blocks that contain any *i*-element set of points.

A **Steiner system** S(t,k,v) is a t-(v,k,λ) design with λ=1. There are no Steiner systems known with t>5. The ones known (to me anyway) for t=5 are as follows:

S(5,6,12), S(5,6,24), S(5,8,24), S(5,7,28), S(5,6,48), S(5,6,72), S(5,6,84),

S(5,6,108), S(5,6,132), S(5,6,168), and S(5,6,244).

**Question**: Are there others with t=5? ANy with $t>5$?

A couple of these are well-known to arise as the support of codewords of a constant weight in a linear code C (as in the Assmus-Mattson theorem, discussed in another post) in the case when C is a Golay code (S(5,6,12) and S(5,8,24)). See also the wikipedia entry for Steiner system.

**Question**: Do any of these others arise “naturally from coding theory” like these two do? I.e., do they all arise as the support of codewords of a constant weight in a linear code C via Assmus-Mattson?

Here is a Sage example to illustrate the case of S(5,8,24):

sage: C = ExtendedBinaryGolayCode()

sage: C.assmus_mattson_designs(5)

[‘weights from C: ‘,

[8, 12, 16, 24],

‘designs from C: ‘,

[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)], [5, (24, 24, 1)]],

‘weights from C*: ‘,

[8, 12, 16],

‘designs from C*: ‘,

[[5, (24, 8, 1)], [5, (24, 12, 48)], [5, (24, 16, 78)]]]

sage: C.assmus_mattson_designs(6)

0

sage: blocks = [c.support() for c in C if hamming_weight(c)==8]; len(blocks)

759

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