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'' $  t $-design,  $  t $-$  ( v, k, \lambda ) $-design on a  $  v $-set  $  S $''

A  $  t $-design is a system of  $  k $-subsets (blocks) of the set  $  S $
such that every  $  t $-subset of elements of  $  S $
appears in exactly  $  \lambda $
blocks. The class of  $  2 $-designs coincides with the class of [[balanced incomplete block design]]s (cf. [[Block design|Block design]]). The name tactical configuration is given to an [[incidence system]] in which every set is incident with exactly  $  k $
elements, while every element is incident with exactly  $  r $
sets. A  $  t $-design with  $  t = k $
is called trivial. If a  $  t $-design is not trivial, then

$$ 
t + 1  \leq   k  \leq   v - 1 - t.
$$

Every  $  t $-design is an  $  s $-design for any  $  s \leq  t $. 
The number  $  \lambda _ {s} $
of occurrences of an arbitrary  $  s $-subset in the blocks of a  $  t $-design is given by the formula

$$ 
\lambda _ {s}  = \ 
\left ( \begin{array}{c}
k - s \\
 t - s 
\end{array}
 \right )  ^ {-1}
\left ( \begin{array}{c}
v - s \\
 t - s 
\end{array}
 \right ) \lambda ,\ \ 
0 < s \leq  t.
$$

The condition that the  $  \lambda _ {s} $
be integers is a necessary condition for the existence of a  $  t $-design. In particular, for  $  t \geq  2 $
every  $  t $-design is a balanced incomplete block design.

The main question concerning  $  t $-designs is the problem of their existence and construction. For a long time, for  $  t > 3 $
only isolated examples were known; in particular  $  5 $-$  ( 12, 6, 1) $- and  $  5 $-$  ( 24, 8, 1) $-designs connected with the five-fold transitive Mathieu groups  $  M _ {12} $
and  $  M _ {24} $ (cf. [[Mathieu group|Mathieu group]]), respectively. However, in the 1960s the connection between  $  t $-designs and coding theory (cf. [[Code|Code]]) was discovered (see, for example, [[#References|[3]]] and [[#References|[4]]]), and, starting from vectors with  $  v $
non-zero coordinates, a way was shown of constructing a  $  t $-design belonging to a linear  $  ( n, k) $-code, which is a  $  k $-dimensional vector subspace of an  $  n $-dimensional space over a [[Finite field|finite field]] (see [[#References|[5]]] and [[#References|[7]]]).

It is well known that  $  t $-fold transitive groups other than the symmetric and the alternating groups lead to non-trivial  $  t $-designs; this yields a few infinite series of  $  3 $-designs. With the help of ideas from group theory and geometry, infinite classes of  $  4 $- and  $  5 $-designs were also constructed (see for example [[#References|[6]]]).

For the number  $  b $
of blocks in a  $  t $-design there is the inequality

$$ \tag{* }
b  \geq  \ 
\left \{

\begin{array}{ll}
\left ( \begin{array}{c}
v \\
 s 
\end{array}
 \right )  & \textrm{ if }  t = 2s, v \geq  k + s,  \\
2 \left ( \begin{array}{c}
v - 1 \\
 s 
\end{array}
 \right )  & \textrm{ if }  t = 2s + 1, v - 1 \geq  k + s,  \\
\end{array}
\right.
$$

generalizing the [[Fisher inequality]]  $  ( b \geq  v) $
for balanced incomplete block designs. In the case of equality in (*), the  $  t $-design is called tight. Tight  $  t $-designs generalize symmetric  $  2 $-designs; in particular, for  $  t = 2s $
the set of intersection numbers of blocks in a tight  $  t $-design contains exactly  $  s $
different elements.

The tight  $  3 $-designs are the Hadamard designs, i.e. they are the  $  3 $-$  ( 4n, 2n, n- 1) $-designs, while for  $  s \geq  2 $
there are no non-trivial tight  $  ( 2s + 1) $-designs. From a given  $  t $-$  ( v, k, \lambda ) $-design three other  $  t $-designs can be obtained: a) taking the complement in  $  S $
of each block; b) discarding an arbitrary element and all the blocks that contain it; and c) taking blocks containing an arbitrarily chosen element and removing it from them. The  $  t $-designs thus obtained are called, respectively, the complementary, the residual and the derived  $  t $-design with respect to the initial one. They are, respectively, a  $  t $-$  ( v, v- k, \lambda  ^ {c} ) $-design with

$$ 
\lambda  ^ {c}  = \ 
\left ( \begin{array}{c}
k \\
 t 
\end{array}
 \right )  ^ {-1}
\left ( \begin{array}{c}
v - k \\
 t 
\end{array}
 \right ) \lambda ,
$$

a  $  ( t - 1) $-$  ( v- 1, k, \lambda  ^ {r} ) $-design with

$$ 
\lambda  ^ {r}  =  ( v - k) ( k - t + 1)  ^ {-1} \lambda
$$

and a  $  ( t - 1) $-$  ( v- 1, k- 1, \lambda ) $-design.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Dembowski,   "Finite geometries" , Springer  (1968)  pp. 254</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.K. Ray-Chaudhuri,   R.M. Wilson,   "On t-designs"  ''Osaka J. Math.'' , '''12'''  (1975)  pp. 737–744</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.F. Assmus,   H.F. Mattson,   "Perfect codes and the Mathieu groups"  ''Arch. Math. Basel'' , '''17'''  (1966)  pp. 121–135</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E.F. Assmus,   H.F. Mattson,   "New 5-designs"  ''J. Comb. Theory'' , '''6'''  (1969)  pp. 122–151</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.V. Semakov,   V.A. Zinov'ev,   "Balanced codes and tactical configurations"  ''Problems Inform. Transmission'' , '''5''' :  3  (1969)  pp. 22–28  ''Probl. Peredachi Inform.'' , '''5''' :  3  (1969)  pp. 28–36</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W.O. Alltop,   "An infiite class of 5-designs"  ''J. Comb. Theory'' , '''12'''  (1972)  pp. 390–395</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V. Pless,   "Symmetry codes over GF(3) and new five-designs"  ''J. Comb. Theory'' , '''12'''  (1972)  pp. 119–142</TD></TR></table>

====Comments====
A  $  t $-$  ( v , k, \lambda ) $-design with  $  \lambda = 1 $
is also known as a [[Steiner system|Steiner system]] and denoted by  $  S( t, k, v ) $; 
an arbitrary  $  t $-$  ( v, k, \lambda ) $-design is sometimes denoted  $  S _  \lambda  ( t, k, v) $.

Of particular interest is the existence of non-trivial  $  t $-designs without repeated blocks (i.e. no  $  k $-subset appears twice in the list of blocks); such  $  t $-designs are called simple. L. Teirlinck [[#References|[a3]]] settled a long-standing conjecture by proving the existence of non-trivial simple  $  t $-designs for every value of  $  t $. 
A list of the known infinite families of simple  $  t $-designs with  $  t \geq  4 $
and a table of simple  $  t $-designs with  $  v \leq  30 $
is given in [[#References|[a4]]].

The only non-trivial tight  $  4 $-design is the unique  $  4 $-$  ( 23, 7, 1) $-design related to the Mathieu group  $  M _ {23} $ (see [[#References|[a5]]]–[[#References|[a7]]]), and there are only finitely many tight  $  2s $-designs for any fixed value  $  s \geq  5 $ (see [[#References|[a8]]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.J. MacWilliams,   N.J.A. Sloane,   "The theory of error-correcting codes" , '''I-II''' , North-Holland  (1978)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  T. Beth,   D. Jungnickel,   H. Lenz,   "Design theory" , Cambridge Univ. Press  (1986)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Teirlinck,   "Non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040110.png" />-designs without repeated blocks exist for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040111.png" />"  ''Disc. Math.'' , '''65'''  (1987)  pp. 301–311</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  Y.M. Chee,   C.J. Colbourn,   D.L. Kreher,   "Simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040112.png" />-designs with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040113.png" />"  ''Ars Comb.'' , '''29'''  (1990)  pp. 193–258</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  N. Ito,   "Tight 4-designs"  ''Osaka J. Math.'' , '''12'''  (1975)  pp. 493–522</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Enomoto,   N. Ito,   R. Noda,   "Tight 4-designs"  ''Osaka J. Math.'' , '''16'''  (1979)  pp. 353–356</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Bremner,   "A diophantine equation arising from tight 4-designs"  ''Osaka J. Math.'' , '''16'''  (1979)  pp. 353–356</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Bannai,   "On tight designs"  ''Quart. J. Math. (Oxford)'' , '''28'''  (1977)  pp. 433–448</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  M. Hall,   "Combinatorial theory" , Blaisdell  (1967)</TD></TR></table>