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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301201.png" /> be a resolvable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301203.png" />-design (see [[Tactical configuration|Tactical configuration]]), that is, the block set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301204.png" /> is partitioned into parallel classes each of which in turn partitions the point set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301205.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301206.png" /> is called affine, or affine resolvable, if there exists a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301207.png" /> such that any two non-parallel blocks intersect in exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301208.png" /> points. For proofs of the results stated below, see [[#References|[a1]]].
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The affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a1301209.png" />-designs are precisely the nets, see [[Net (in finite geometry)|Net (in finite geometry)]], and the affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012010.png" />-designs coincide with the Hadamard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012012.png" />-designs, that is, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012013.png" />-designs, cf. [[Tactical configuration|Tactical configuration]]. There are no non-trivial affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012014.png" />-designs with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012015.png" />. Thus, the most interesting case is that of affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012016.png" />-designs, which are often simply called affine designs. Any affine <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012017.png" />-design satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012019.png" /> denotes the number of blocks through a point and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012020.png" /> denotes the number of blocks in a parallel class. Moreover, equality holds in this inequality if and only the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012021.png" />-design is an (affine) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012022.png" />-design. Any resolvable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012023.png" />-design satisfies the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012024.png" />, and equality holds if and only the design is affine. In this case, all parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012025.png" /> may be written in terms of the two parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012027.png" />, as follows:
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Let $\mathcal{D} = ( V , \mathcal{B} )$ be a resolvable $t - ( v , k , \lambda )$-design (see [[Tactical configuration|Tactical configuration]]), that is, the block set of $\mathcal{D}$ is partitioned into parallel classes each of which in turn partitions the point set $V$. $\mathcal{D}$ is called affine, or affine resolvable, if there exists a constant $\mu$ such that any two non-parallel blocks intersect in exactly $\mu$ points. For proofs of the results stated below, see [[#References|[a1]]].
  
and the design is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012029.png" />.
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The affine $1$-designs are precisely the nets, see [[Net (in finite geometry)|Net (in finite geometry)]], and the affine $3$-designs coincide with the Hadamard $3$-designs, that is, the $3 - ( 4 \mu , 2 \mu , \mu - 1 )$-designs, cf. [[Tactical configuration|Tactical configuration]]. There are no non-trivial affine $t$-designs with $t \geq 4$. Thus, the most interesting case is that of affine $2$-designs, which are often simply called affine designs. Any affine $1$-design satisfies the inequality $r \leq ( s ^ { 2 } \mu - 1 ) / ( \mu - 1 )$, where $r$ denotes the number of blocks through a point and where $s$ denotes the number of blocks in a parallel class. Moreover, equality holds in this inequality if and only the $1$-design is an (affine) $2$-design. Any resolvable $2$-design satisfies the inequality $r \geq k + \lambda$, and equality holds if and only the design is affine. In this case, all parameters of $\mathcal{D}$ may be written in terms of the two parameters $s$ and $\mu$, as follows:
  
The outstanding problem in this area is to characterize the possible pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012030.png" /> for which an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012031.png" /> exists. The only known pairs to date (2001) are those with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012032.png" /> and the pairs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012033.png" /> for some prime power <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012034.png" /> and some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012035.png" />. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012036.png" /> corresponds to Hadamard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012038.png" />-designs, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012039.png" />-designs; any such design extends uniquely to a Hadamard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012040.png" />-design, and existence — which is equivalent to that of an [[Hadamard matrix|Hadamard matrix]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012041.png" /> — is conjectured for all values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012042.png" />. The classical examples for the second case are the affine designs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012043.png" /> formed by the points and hyperplanes of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012044.png" />-dimensional finite affine spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012045.png" /> over the [[Galois field|Galois field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012046.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012047.png" /> (so <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012048.png" /> is a prime power here; cf. also [[Affine space|Affine space]]). As to the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012049.png" />, a design <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012050.png" /> is just an affine plane of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012051.png" />, see also [[Plane|Plane]].
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\begin{equation*} k = s \mu , v = s ^ { 2 } \mu , \lambda = \frac { s \mu - 1 } { \mu - 1 } , r = \frac { s ^ { 2 } \mu - 1 } { \mu - 1 }, \end{equation*}
  
In general, an affine design cannot be characterized just by its parameters. For instance, the number of non-isomorphic designs with the same parameters as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012052.png" /> grows exponentially with a growth rate of at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012054.png" />. Hence, it is desirable to characterize the designs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012055.png" /> among the affine or resolvable designs. For instance, by Dembowski's theorem, a resolvable design <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012056.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012058.png" /> in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is isomorphic to some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012059.png" />; the same conclusion holds if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012060.png" /> admits an automorphism group which is transitive on ordered triples of non-collinear points. See [[#References|[a1]]], Sec. XII.3, for proofs and further characterizations. In particular, there is a wealth of results characterizing the classical affine planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130120/a13012061.png" /> and other interesting classes of affine planes; for example, a result of Y. Hiramine [[#References|[a2]]] states that any finite affine plane that admits a collineation group acting primitively on points is a translation plane (cf. [[Plane|Plane]]; [[Primitive group of permutations|Primitive group of permutations]]). Detailed studies of translation planes may be found in [[#References|[a3]]] and [[#References|[a4]]].
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and the design is denoted by $A _ { \mu } ( s )$.
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The outstanding problem in this area is to characterize the possible pairs $( s , \mu )$ for which an $A _ { \mu } ( s )$ exists. The only known pairs to date (2001) are those with $s = 2$ and the pairs of the form $( q , q ^ { d - 2 } )$ for some prime power $q$ and some integer $d \geq 2$. The case $s = 2$ corresponds to Hadamard $2$-designs, i.e. $2 - ( 4 \mu - 1,2 \mu - 1 , \mu - 1 )$-designs; any such design extends uniquely to a Hadamard $3$-design, and existence — which is equivalent to that of an [[Hadamard matrix|Hadamard matrix]] of order $4 \mu$ — is conjectured for all values of $\mu$. The classical examples for the second case are the affine designs $A G _ { d -1}  ( d , q )$ formed by the points and hyperplanes of the $d$-dimensional finite affine spaces $A G ( d , q )$ over the [[Galois field|Galois field]] $\operatorname {GF} ( q )$ of order $q$ (so $q$ is a prime power here; cf. also [[Affine space|Affine space]]). As to the case $d = 2$, a design $A _ { 1 } ( s )$ is just an affine plane of order $s$, see also [[Plane|Plane]].
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In general, an affine design cannot be characterized just by its parameters. For instance, the number of non-isomorphic designs with the same parameters as $A G _ { d -1}  ( d , q )$ grows exponentially with a growth rate of at least $e ^ { k .\operatorname { ln } k }$, where $k = q ^ { d - 1 }$. Hence, it is desirable to characterize the designs $A G _ { d -1}  ( d , q )$ among the affine or resolvable designs. For instance, by Dembowski's theorem, a resolvable design $\mathcal{D}$ with $\lambda &gt; 1$ and $s &gt; 2$ in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is isomorphic to some $A G _ { d -1}  ( d , q )$; the same conclusion holds if $\mathcal{D}$ admits an automorphism group which is transitive on ordered triples of non-collinear points. See [[#References|[a1]]], Sec. XII.3, for proofs and further characterizations. In particular, there is a wealth of results characterizing the classical affine planes $A G ( 2 , q )$ and other interesting classes of affine planes; for example, a result of Y. Hiramine [[#References|[a2]]] states that any finite affine plane that admits a collineation group acting primitively on points is a translation plane (cf. [[Plane|Plane]]; [[Primitive group of permutations|Primitive group of permutations]]). Detailed studies of translation planes may be found in [[#References|[a3]]] and [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Beth,  D. Jungnickel,  H. Lenz,  "Design theory" , Cambridge Univ. Press  (1999)  (Edition: Second)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Y. Hiramine,  "Affine planes with primitive collineation groups"  ''J. Algebra'' , '''128'''  (1990)  pp. 366–383</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M.J. Kallaher,  "Affine planes with transitive collineation groups" , North-Holland  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Lüneburg,  "Translation planes" , Springer  (1980)</TD></TR></table>
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<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  T. Beth,  D. Jungnickel,  H. Lenz,  "Design theory" , Cambridge Univ. Press  (1999)  (Edition: Second)</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  Y. Hiramine,  "Affine planes with primitive collineation groups"  ''J. Algebra'' , '''128'''  (1990)  pp. 366–383</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top">  M.J. Kallaher,  "Affine planes with transitive collineation groups" , North-Holland  (1981)</td></tr>
 +
<tr><td valign="top">[a4]</td> <td valign="top">  H. Lüneburg,  "Translation planes" , Springer  (1980) {{ZBL|0446.51003}}</td></tr></table>

Latest revision as of 11:00, 17 March 2023

Let $\mathcal{D} = ( V , \mathcal{B} )$ be a resolvable $t - ( v , k , \lambda )$-design (see Tactical configuration), that is, the block set of $\mathcal{D}$ is partitioned into parallel classes each of which in turn partitions the point set $V$. $\mathcal{D}$ is called affine, or affine resolvable, if there exists a constant $\mu$ such that any two non-parallel blocks intersect in exactly $\mu$ points. For proofs of the results stated below, see [a1].

The affine $1$-designs are precisely the nets, see Net (in finite geometry), and the affine $3$-designs coincide with the Hadamard $3$-designs, that is, the $3 - ( 4 \mu , 2 \mu , \mu - 1 )$-designs, cf. Tactical configuration. There are no non-trivial affine $t$-designs with $t \geq 4$. Thus, the most interesting case is that of affine $2$-designs, which are often simply called affine designs. Any affine $1$-design satisfies the inequality $r \leq ( s ^ { 2 } \mu - 1 ) / ( \mu - 1 )$, where $r$ denotes the number of blocks through a point and where $s$ denotes the number of blocks in a parallel class. Moreover, equality holds in this inequality if and only the $1$-design is an (affine) $2$-design. Any resolvable $2$-design satisfies the inequality $r \geq k + \lambda$, and equality holds if and only the design is affine. In this case, all parameters of $\mathcal{D}$ may be written in terms of the two parameters $s$ and $\mu$, as follows:

\begin{equation*} k = s \mu , v = s ^ { 2 } \mu , \lambda = \frac { s \mu - 1 } { \mu - 1 } , r = \frac { s ^ { 2 } \mu - 1 } { \mu - 1 }, \end{equation*}

and the design is denoted by $A _ { \mu } ( s )$.

The outstanding problem in this area is to characterize the possible pairs $( s , \mu )$ for which an $A _ { \mu } ( s )$ exists. The only known pairs to date (2001) are those with $s = 2$ and the pairs of the form $( q , q ^ { d - 2 } )$ for some prime power $q$ and some integer $d \geq 2$. The case $s = 2$ corresponds to Hadamard $2$-designs, i.e. $2 - ( 4 \mu - 1,2 \mu - 1 , \mu - 1 )$-designs; any such design extends uniquely to a Hadamard $3$-design, and existence — which is equivalent to that of an Hadamard matrix of order $4 \mu$ — is conjectured for all values of $\mu$. The classical examples for the second case are the affine designs $A G _ { d -1} ( d , q )$ formed by the points and hyperplanes of the $d$-dimensional finite affine spaces $A G ( d , q )$ over the Galois field $\operatorname {GF} ( q )$ of order $q$ (so $q$ is a prime power here; cf. also Affine space). As to the case $d = 2$, a design $A _ { 1 } ( s )$ is just an affine plane of order $s$, see also Plane.

In general, an affine design cannot be characterized just by its parameters. For instance, the number of non-isomorphic designs with the same parameters as $A G _ { d -1} ( d , q )$ grows exponentially with a growth rate of at least $e ^ { k .\operatorname { ln } k }$, where $k = q ^ { d - 1 }$. Hence, it is desirable to characterize the designs $A G _ { d -1} ( d , q )$ among the affine or resolvable designs. For instance, by Dembowski's theorem, a resolvable design $\mathcal{D}$ with $\lambda > 1$ and $s > 2$ in which every line (that is, the intersection of all blocks through two given points) meets every non-parallel block is isomorphic to some $A G _ { d -1} ( d , q )$; the same conclusion holds if $\mathcal{D}$ admits an automorphism group which is transitive on ordered triples of non-collinear points. See [a1], Sec. XII.3, for proofs and further characterizations. In particular, there is a wealth of results characterizing the classical affine planes $A G ( 2 , q )$ and other interesting classes of affine planes; for example, a result of Y. Hiramine [a2] states that any finite affine plane that admits a collineation group acting primitively on points is a translation plane (cf. Plane; Primitive group of permutations). Detailed studies of translation planes may be found in [a3] and [a4].

References

[a1] T. Beth, D. Jungnickel, H. Lenz, "Design theory" , Cambridge Univ. Press (1999) (Edition: Second)
[a2] Y. Hiramine, "Affine planes with primitive collineation groups" J. Algebra , 128 (1990) pp. 366–383
[a3] M.J. Kallaher, "Affine planes with transitive collineation groups" , North-Holland (1981)
[a4] H. Lüneburg, "Translation planes" , Springer (1980) Zbl 0446.51003
How to Cite This Entry:
Affine design. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Affine_design&oldid=15782
This article was adapted from an original article by Dieter Jungnickel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article