View source for Tactical configuration

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t0920402.png" />-design, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t0920405.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t0920406.png" />-design on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t0920407.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t0920408.png" />''

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t0920409.png" />-design is a system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204010.png" />-subsets (blocks) of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204011.png" /> such that every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204012.png" />-subset of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204013.png" /> appears in exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204014.png" /> blocks. The class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204015.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204016.png" /> elements, while every element is incident with exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204017.png" /> sets. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204018.png" />-design with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204019.png" /> is called trivial. If a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204021.png" />-design is not trivial, then

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Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204023.png" />-design is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204024.png" />-design for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204025.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204026.png" /> of occurrences of an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204027.png" />-subset in the blocks of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204028.png" />-design is given by the formula

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The condition that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204030.png" /> be integers is a necessary condition for the existence of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204031.png" />-design. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204032.png" /> every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204033.png" />-design is a balanced incomplete block design.

The main question concerning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204034.png" />-designs is the problem of their existence and construction. For a long time, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204035.png" /> only isolated examples were known; in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204036.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204037.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204038.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204039.png" />-designs connected with the five-fold transitive Mathieu groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204041.png" /> (cf. [[Mathieu group|Mathieu group]]), respectively. However, in the 1960s the connection between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204042.png" />-designs and coding theory (cf. [[Code|Code]]) was discovered (see, for example, [[#References|[3]]] and [[#References|[4]]]), and, starting from vectors with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204043.png" /> non-zero coordinates, a way was shown of constructing a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204044.png" />-design belonging to a linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204046.png" />-code, which is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204047.png" />-dimensional vector subspace of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204048.png" />-dimensional space over a [[Finite field|finite field]] (see [[#References|[5]]] and [[#References|[7]]]).

It is well known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204049.png" />-fold transitive groups other than the symmetric and the alternating groups lead to non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204050.png" />-designs; this yields a few infinite series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204051.png" />-designs. With the help of ideas from group theory and geometry, infinite classes of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204052.png" />- and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204053.png" />-designs were also constructed (see for example [[#References|[6]]]).

For the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204054.png" /> of blocks in a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204055.png" />-design there is the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

generalizing the [[Fisher inequality]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204057.png" /> for balanced incomplete block designs. In the case of equality in (*), the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204058.png" />-design is called tight. Tight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204060.png" />-designs generalize symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204061.png" />-designs; in particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204062.png" /> the set of intersection numbers of blocks in a tight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204063.png" />-design contains exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204064.png" /> different elements.

The tight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204065.png" />-designs are the Hadamard designs, i.e. they are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204066.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204067.png" />-designs, while for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204068.png" /> there are no non-trivial tight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204069.png" />-designs. From a given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204070.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204071.png" />-design three other <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204072.png" />-designs can be obtained: a) taking the complement in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204073.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204074.png" />-designs thus obtained are called, respectively, the complementary, the residual and the derived <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204078.png" />-design with respect to the initial one. They are, respectively, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204079.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204080.png" />-design with

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a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204082.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204083.png" />-design with

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204084.png" /></td> </tr></table>

and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204085.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204086.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204087.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204088.png" />-design with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204089.png" /> is also known as a [[Steiner system|Steiner system]] and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204090.png" />; an arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204091.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204092.png" />-design is sometimes denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204093.png" />.

Of particular interest is the existence of non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204094.png" />-designs without repeated blocks (i.e. no <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204095.png" />-subset appears twice in the list of blocks); such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204096.png" />-designs are called simple. L. Teirlinck [[#References|[a3]]] settled a long-standing conjecture by proving the existence of non-trivial simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204098.png" />-designs for every value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t09204099.png" />. A list of the known infinite families of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040100.png" />-designs with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040101.png" /> and a table of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040102.png" />-designs with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040103.png" /> is given in [[#References|[a4]]].

The only non-trivial tight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040104.png" />-design is the unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040105.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040106.png" />-design related to the Mathieu group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040107.png" /> (see [[#References|[a5]]]–[[#References|[a7]]]), and there are only finitely many tight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040108.png" />-designs for any fixed value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092040/t092040109.png" /> (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>