incidence structure IncidenceStructures 2005-04-08 14:08:27 2008-12-21 13:50:55 Topic design block block design $\tau$-design simple design square design symmetric design tactical configuration balanced incomplete block design BIBD Steiner system Steiner triple system projective plane finite projective plane affine plane finite affine plane incidence block design \usepackage{amssymb} % \usepackage{amsmath} % \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % portions from % makra.sty 1989-2005 by Marijke van Gans % % ^ ^ \catcode`\@=11 % o o % ->*<- % ~ %%%% CHARS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % code char frees for \let\Para\S % \Para § \S \scriptstyle \let\Pilcrow\P % \Pilcrow ¶ \P \mathchardef\pilcrow="227B \mathchardef\lt="313C % \lt < < bra \mathchardef\gt="313E % \gt > > ket \let\bs\backslash % \bs \ \let\us\_ % \us _ \_ ... %%%% DIACRITICS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %let\udot\d % under-dot (text mode), frees \d \let\odot\. % over-dot (text mode), frees \. %let\hacek\v % hacek (text mode), frees \v %let\makron\= % makron (text mode), frees \= %let\tilda\~ % tilde (text mode), frees \~ \let\uml\" % umlaut (text mode), frees \" %def\ij/{i{\kern-.07em}j} %def\trema#1{\discretionary{-}{#1}{\uml #1}} %%%% amssymb %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\le\leqslant \let\ge\geqslant %let\prece\preceqslant %let\succe\succeqslant %%%% USEFUL MISC %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %def\C++{C$^{_{++}}$} %let\writelog\wlog %def\wl@g/{{\sc wlog}} %def\wlog{\@ifnextchar/{\wl@g}{\writelog}} %def\org#1{\lower1.2pt\hbox{#1}} % chem struct formulae: \bs, --- / \org{C} etc. %%%% USEFUL INTERNAL LaTeX STUFF %%%%%%%%%%%%%%%%%%%%%%%% %let\Ifnextchar=\@ifnextchar %let\Ifstar=\@ifstar %def\currsize{\@currsize} %%%% KERNING, SPACING, BREAKING %%%%%%%%%%%%%%%%%%%%%%%%% \def\comma{,\,\allowbreak} %def\qqquad{\hskip3em\relax} %def\qqqquad{\hskip4em\relax} %def\qqqqquad{\hskip5em\relax} %def\qqqqqquad{\hskip6em\relax} %def\qqqqqqquad{\hskip7em\relax} %def\qqqqqqqquad{\hskip8em\relax} %%%% LAYOUT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% COUNTERS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %let\addtoreset\@addtoreset %{A}{B} adds A to list of counters reset to 0 % when B is \refstepcounter'ed (see latex.tex) % %def\numbernext#1#2{\setcounter{#1}{#2}\addtocounter{#1}{\m@ne}} %%%% EQUATIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% LEMMATA %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% DISPLAY %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% MATH LAYOUT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \let\D\displaystyle \let\T\textstyle \let\S\scriptstyle \let\SS\scriptscriptstyle % array: %def\<#1:{\begin{array}{#1}} %def\>{\end{array}} % array using [ ] with rounded corners: %def\[#1:{\left\lgroup\begin{array}{#1}} %def\]{\end{array}\right\rgroup} % array using ( ): %def\(#1:{\left(\begin{array}{#1}} %def\){\end{array}\right)} %def\hh{\noalign{\vskip\doublerulesep}} %%%% MATH SYMBOLS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %def\d{\mathord{\rm d}} % d as in dx %def\e{{\rm e}} % e as in e^x %def\Ell{\hbox{\it\char`\$}} \def\sfmath#1{{\mathchoice% {{\sf #1}}{{\sf #1}}{{\S\sf #1}}{{\SS\sf #1}}}} \def\Stalkset#1{\sfmath{I\kern-.12em#1}} \def\Bset{\Stalkset B} \def\Nset{\Stalkset N} \def\Rset{\Stalkset R} \def\Hset{\Stalkset H} \def\Fset{\Stalkset F} \def\kset{\Stalkset k} \def\In@set{\raise.14ex\hbox{\i}\kern-.237em\raise.43ex\hbox{\i}} \def\Roundset#1{\sfmath{\kern.14em\In@set\kern-.4em#1}} \def\Qset{\Roundset Q} \def\Cset{\Roundset C} \def\Oset{\Roundset O} \def\Zset{\sfmath{Z\kern-.44emZ}} % \frac overwrites LaTeX's one (use TeX \over instead) %def\fraq#1#2{{}^{#1}\!/\!{}_{\,#2}} \def\frac#1#2{\mathord{\mathchoice% {\T{#1\over#2}} {\T{#1\over#2}} {\S{#1\over#2}} {\SS{#1\over#2}}}} %def\half{\frac12} \mathcode`\<="4268 % < now is \langle, \lt is < \mathcode`\>="5269 % > now is \rangle, \gt is > %def\biggg#1{{\hbox{$\left#1\vbox %to20.5\p@{}\right.\n@space$}}} %def\Biggg#1{{\hbox{$\left#1\vbox %to23.5\p@{}\right.\n@space$}}} \let\epsi=\varepsilon \def\omikron{o} \def\Alpha{{\rm A}} \def\Beta{{\rm B}} \def\Epsilon{{\rm E}} \def\Zeta{{\rm Z}} \def\Eta{{\rm H}} \def\Iota{{\rm I}} \def\Kappa{{\rm K}} \def\Mu{{\rm M}} \def\Nu{{\rm N}} \def\Omikron{{\rm O}} \def\Rho{{\rm P}} \def\Tau{{\rm T}} \def\Ypsilon{{\rm Y}} % differs from \Upsilon \def\Chi{{\rm X}} %def\dg{^{\circ}} % degrees %def\1{^{-1}} % inverse \def\*#1{{\bf #1}} % boldface e.g. vector %def\vi{\mathord{\hbox{\bf\i}}} % boldface vector \i %def\vj{\mathord{\,\hbox{\bf\j}}} % boldface vector \j %def\union{\mathbin\cup} %def\isect{\mathbin\cap} %let\so\Longrightarrow %let\oso\Longleftrightarrow %let\os\Longleftarrow % := and :<=> %def\isdef{\mathrel{\smash{\stackrel{\SS\rm def}{=}}}} %def\iffdef{\mathrel{\smash{stackrel{\SS\rm def}{\oso}}}} \def\isdef{\mathrel{\mathop{=}\limits^{\smash{\hbox{\tiny def}}}}} %def\iffdef{\mathrel{\mathop{\oso}\limits^{\smash{\hbox{\tiny %def}}}}} %def\tr{\mathop{\rm tr}} % tr[ace] %def\ter#1{\mathop{^#1\rm ter}} % k-ter[minant] %let\.=\cdot %let\x=\times % ח (direct product) %def\qed{ ${\S\circ}\!{}^\circ\!{\S\circ}$} %def\qed{\vrule height 6pt width 6pt depth 0pt} %def\edots{\mathinner{\mkern1mu % \raise7pt\vbox{\kern7pt\hbox{.}}\mkern1mu % .shorter % \raise4pt\hbox{.}\mkern1mu % . % \raise1pt\hbox{.}\mkern1mu}} % . %def\fdots{\mathinner{\mkern1mu % \raise7pt\vbox{\kern7pt\hbox{.}} % . ~45° % \raise4pt\hbox{.} % . % \raise1pt\hbox{.}\mkern1mu}} % . \def\mod#1{\allowbreak \mkern 10mu({\rm mod}\,\,#1)} % redefines TeX's one using less space %def\int{\intop\displaylimits} %def\oint{\ointop\displaylimits} %def\intoi{\int_0^1} %def\intall{\int_{-\infty}^\infty} %def\su#1{\mathop{\sum\raise0.7pt\hbox{$\S\!\!\!\!\!#1\,$}}} %let\frakR\Re %let\frakI\Im %def\Re{\mathop{\rm Re}\nolimits} %def\Im{\mathop{\rm Im}\nolimits} %def\conj#1{\overline{#1\vphantom1}} %def\cj#1{\overline{#1\vphantom+}} %def\forAll{\mathop\forall\limits} %def\Exists{\mathop\exists\limits} %%%% PICTURES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %def\cent{\makebox(0,0)} %def\node{\circle*4} %def\nOde{\circle4} %%%% REFERENCES %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %def\opcit{[{\it op.\,cit.}]} \def\bitem#1{\bibitem[#1]{#1}} \def\name#1{{\sc #1}} \def\book#1{{\sl #1\/}} \def\paper#1{``#1''} \def\mag#1{{\it #1\/}} \def\vol#1{{\bf #1}} \def\isbn#1{{\small\tt ISBN\,\,#1}} \def\seq#1{{\small\tt #1}} %def\url<{\verb>} %def\@cite#1#2{[{#1\if@tempswa\ #2\fi}]} %%%% VERBATIM CODE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %def\"{\verb"} %%%% AD HOC %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\Pow{\mathop{\rm Pow}\nolimits} \def\I{{\cal I}} \def\P{{\cal P}} \def\B{{\cal B}} \def\0#1{\hbox{\sc #1}} \def\Bp{\B_{\SS\rm P}} \def\Pb{\P_{\SS\rm B}} \def\Pbi{\P_{{\SS\rm B}'}} \def\Pbii{\P_{{\SS\rm B}''}} %%%% WORDS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \hyphenation{pre-sent pre-sents pre-sent-ed pre-sent-ing % re-pre-sent re-pre-sents re-pre-sent-ed re-pre-sent-ing % re-fer-ence re-fer-ences re-fer-enced re-fer-encing % ge-o-met-ry re-la-ti-vi-ty Gauss-ian Gauss-ians % Des-ar-gues-ian} %def\oord/{o{\trema o}rdin\-ate} % usage: C\oord/s, c\oord/. % output: co\"ord... except when linebreak at co-ord... %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ^ ^ \catcode`\@=12 % ` ' % ->*<- % ~ \PMlinkescapeword{alphabets} \PMlinkescapeword{block} \PMlinkescapeword{blocks} \PMlinkescapeword{combinations} \PMlinkescapeword{constant} \PMlinkescapeword{decompositions} \PMlinkescapeword{difference} \PMlinkescapeword{incident} \PMlinkescapeword{incomplete} \PMlinkescapeword{meet} \PMlinkescapeword{order} \PMlinkescapeword{orders} \PMlinkescapeword{property} \PMlinkescapeword{restricted} \PMlinkescapeword{satisfies} \PMlinkescapeword{simple} \PMlinkescapeword{size} \PMlinkescapeword{square} \PMlinkescapeword{states} \PMlinkescapeword{structure} \PMlinkescapeword{term} \PMlinkescapeword{type} Let $\P$ and $\B$ be two disjoint sets; the elements of $\P$ will be termed {\bf points} and those of $\B$ {\bf blocks}. Now (throughout this article, a bullet point defines the term in boldface) % \begin{itemize} \item an {\bf incidence structure} is a subset $\I\subseteq\P\times\B$ \end{itemize} % and a point $\0p$ and a block $\0b$ are said to be {\bf incident} iff $(\0p,\0b)\in\I$. The {\bf dual} incidence structure $\I^*$ is the same structure with the labels ``point'' and ``block'' reversed. Every block $\0b$ has a set $\Pb\subseteq \P$ of points it is incident with. If $\Pbi\ne\Pbii$ whenever $\0b' \ne \0b''$, the incidence structure is said to be {\bf simple}. Now we could identify each block $\0b$ with its $\Pb$ so that blocks no longer {\em have\/} sets of points they are incident with but {\em are\/} such sets. If we define it that way, then % \begin{itemize} \item[$\circ$] a {\bf simple} incidence structure consists of a set $\P$ and a set $\B\subseteq P(\P)$ where $P(\P)$ is the powerset of $\P$ (the set of subsets of $\P$). \end{itemize} % A simple incidence structure is also called a hypergraph (with points as vertices, and blocks as an extended type of ``edges'' that are no longer restricted to exactly two vertices each). Every point $\0p$ also has a set $\Bp\subseteq \B$ of blocks it is incident with. Often, a simple incidence structure also has a simple dual, but the set theory formalism does not allow us to regard blocks as sets of points and simultaneously points as sets of blocks! Nevertheless, it is often useful to alternate between these dual interpretations. \clearpage \subsection*{Designs} \begin{itemize} \item A $\tau$-$(\nu,\kappa,\lambda)$ {\bf design}, aka $\tau${\bf-design} or {\bf block design}, is an incidence structure with \begin{itemize} \item $|\P| = \nu$ points in all, \item $|\Pb| = \kappa$ points in each block $\0b$, and such that \item any set $\0t\subseteq \P$ of $|\0t|=\tau$ points occurs as subset $\0t\subseteq \Pb$ in exactly $\lambda$ blocks. \end{itemize} % \end{itemize} % The parameters are often called $t$, $v$, $k$, $\lambda$ (in mixed Latin and Greek alphabets). Note there may be several non-isomorphic designs with the same values of the parameters, and no designs at all for certain combinations of values. Designs need not be simple (they can have {\bf repeated blocks}), but they usually are (and don't) in which case $\0b$ can again be used as synonym for $\Pb$. % \begin{itemize} \item[$\circ$] 0-designs ($\tau=0$) are allowed. \item[$\circ$] 1-designs ($\tau=1$) are known as {\bf tactical configurations}. \item[$\circ$] 2-designs are called {\bf balanced incomplete block designs} or {\bf BIBD}. \item[$\circ$] 3, 4, 5\dots\ -designs have all been studied. \end{itemize} % Being a $\tau$-$(\nu,\kappa,\lambda)$ design implies also being an $\iota$-$(\nu,\kappa,\lambda_{\,\iota})$ design for every $0\le\iota\le\tau$ (on the same $\nu$ points and with the same block size $\kappa$), with $\lambda_{\,\iota}$ given by $\lambda_{\,\tau}=\lambda$ and recursively $$ \lambda_{\,\iota} \,=\; {\nu-\iota \over \kappa-\iota} \,\lambda_{\,\iota+1} $$ from which we get the number of blocks as $$ \lambda_0 \,=\; {\nu!\,/\,(\nu-\tau)! \over \kappa!\,/\,(\kappa-\tau)!} \;=\; {\nu\choose\tau} \bigg/ {\kappa\choose\tau} $$ Being a 0-design says nothing more than all blocks having the same size. As soon as we have $\tau\ge1$ however we also have a 1-design, so the number $\lambda_1 = |\Bp|$ of blocks per point is constant throughout the structure. Note now $$ \lambda_0\,\kappa \;=\; \lambda_1\,\nu $$ which is also evident from their interpretation. \clearpage As an example: designs (simple designs) with $\kappa=2$ are multigraphs (simple {\bf graphs}), now % \begin{itemize} \item[$\circ$] $\tau=0$ implies no more than that, \item[$\circ$] $\tau=1$ gives {\bf regular graphs}, and \item[$\circ$] $\tau=2$ gives {\bf complete graphs}. \end{itemize} % A more elaborate ``lambda calculus'' (pun intended) can be introduced as follows. Let $\Iota\subseteq\P$ and $\Omikron\subseteq\P$ with $|\Iota|=\iota$ and $|\Omikron|=\omikron$. The number of blocks $\0b$ such that all the points of $\Iota$ are inside $\0b$ and all the points of $\Omikron$ are outside $\0b$ is independent of the choice of $\Iota$ and $\Omikron$, only depending on $\iota$ and $\omikron$, provided $\iota+\omikron\le\tau$. Call this number $\lambda_\iota^\omikron$. It satisfies a kind of reverse Pascal triangle like recursion $$ \lambda_\iota^\omikron \,=\, \lambda_{\iota+1}^\omikron + \lambda_\iota^{\omikron+1} $$ that starts off for $\omikron=0$ with $\lambda_\iota^0 = \lambda_\iota$. An important quantity (for designs with $\tau\ge2$) is the {\bf order} $\lambda_1^1 = \lambda_1^0 - \lambda_2^0 = \lambda_1 - \lambda_2$. Finally, the dual of a design can be a design but need not be. % \begin{itemize} \item A {\bf square design} aka {\bf symmetric design} is one where $\tau=2$ and $|\P|=|\B|$, now also $|\Pb|=|\Bp|$. Here the dual is also a square design. \end{itemize} % Note that for $\tau\ge3$ no designs exist with $|\P|=|\B|$ other than trivial ones (where any $\kappa=\nu-1$ points form a block). \clearpage \subsection*{Steiner systems} \begin{itemize} \item An $S(\tau,\kappa,\nu)$ {\bf Steiner system} is a $\tau$-$(\nu,\kappa,1)$ design (i.e.\ $\lambda=1$). \end{itemize} % Again, there may be several non-isomorphic systems with the same values of the parameters, and no systems at all for certain combinations of values. Note that $\lambda=1$ implies a simple incidence structure; from now on we will interpret a block to {\em be\/} a set of points ($\0b=\Pb$). Let ${\cal S}$ be an $S(\tau,\kappa,\nu)$ with point set $\P$ and block set $\B$, and choose a point $\0p_0\in \P$ (often, the system is so symmetric that it makes no difference which point you choose). The choice uniquely induces an $S({\tau-1}\comma{\kappa-1}\comma{\nu-1})$ ${\cal S}_1$ with point set $\P_1 = \P\setminus\{\0p_0\}$ and block set $\B_1$ consisting of $\0b\setminus\{\0p_0\}$ for only those $\0b\in\B$ that contained $\0p_0$. This works because for any $\0t_1\subseteq\P_1$ with $|\0t_1|=\tau-1$ there was a unique $\0b\in\B$ that contained $\0t=\0t_1\cup\{\0p_0\}$. This recurses down all the way to $\tau=1$ (a partition of $\nu-\tau+1$ into blocks of $\kappa-\tau+1$) and finally to $\tau=0$ (one arbitrary block of $\kappa-\tau$). If any of the divisibility conditions on the way there do not hold, there cannot exist a Steiner system with the original parameters either. For instance, {\bf Steiner triple systems} $S(2,3,\nu)$ (the first Steiner systems studied, by Kirkman, before Steiner) exist for $\nu=0$ and all $\nu\equiv1$ or $3\mod6$, and no other $\nu$. The reverse construction, turning an $S(\tau,\kappa,\nu)$ into an $S({\tau+1}\comma {\kappa+1}\comma {\nu+1})$, need not be unique and may be impossible. Famously an $S(4,5,11)$ and a $S(5,6,12)$ have the Mathieu groups $M_{11}$ and $M_{12}$ as their automorphism groups, while $M_{22}$, $M_{23}$ and $M_{24}$ are those of an $S(3,6,22)$, $S(4,7,23)$ and $S(5,8,24)$, with connexions to the binary Golay code and the Leech lattice. \clearpage \subsection*{Finite planes} The term {\bf line} has a specific meaning for 2-designs in general: for any two points, it is the intersection of all blocks containing both those points. For 2-designs that are also Steiner systems ($\tau=2$ and $\lambda=1$) there is only one such block, so line becomes a synonym for block. And it becomes a finite analogue of the usual geometric meaning of the word. % \begin{itemize} \item An $S(2,\kappa,\nu)$ is the finite analogue of a {\bf plane}, with blocks in the r\^ole of {\bf lines} \end{itemize} % in the following sense: the design property now requires there to be, for any two different points, exactly one line ``through'' both those points. Just like in a real (continuous) plane. This also implies that, for any two different lines $l$ and $m$, there is {\em no more\/} than one point ``on'' both those lines (if both of $\0p$ and $\0q$ were on both those lines, there would be two lines through those points). It does not imply there is always such a point: just like in a real plane, lines can be parallel. One example is a (finite) {\bf affine plane} with $q^2$ points and $q^2+q$ lines. It can be obtained by deleting one line (and all its points) from a projective plane (for which see below). Lines that used to intersect in one of the deleted points are parallel in the affine plane. % \begin{itemize} \item A (finite) {\bf projective plane} is an $S(2\comma q+1\comma q^2+q+1)$ \end{itemize} % and it has no parallel lines. Because any two lines meet in a point, the dual is again a projective plane. So a projective plane is a square design, as well as being a great many other things. It is easy to prove that the property of being a plane dual to a plane (i.e.\ the absence of parallel lines) implies, apart from a few trivial cases, numbers of the form $q+1$ and $q^2+q+1$. Much harder is determining for which $q$ such planes exist. The parameter $q$ is known as the {\bf order} of the plane (this agrees with order as defined above for designs in general). Highly symmetric ``classical'' (aka Desarguesian, Pappian) projective planes can be constructed based on finite fields, for any prime power $q$. Many non-Desarguesian projective planes are known, but thus far their $q$ are also prime powers. The {\bf prime power conjecture} is that orders of all projective planes will be prime powers. The {\bf Bruck--Ryser theorem} states that if $q\equiv1$ or $2\mod4$, and not (a square or) the sum of two squares, it cannot be the order of a projective plane. This rules out 6 for instance, as well as 14 etc. It has been extended to the {\bf Bruck--Ryser--Chowla theorem} for all square 2-designs, with a more complicated constraint. The only other order ruled out to date is 10, via an epic computer search by Lam, Swiercz and Thiel (read {\tt\PMlinkexternal{http://www.cecm.sfu.ca/organics/papers/lam/index.html}{http://www.cecm.sfu.ca/organics/papers/lam/index.html}} for Lam's account). \vfill\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \raggedright \begin{thebibliography}{MST73} \bitem{AK93} \name{E.\,F.\,Assmus} and \name{J.\,D.\,Key}, \book{Designs and their Codes}\\ (pbk.~ed.~w.~corr.), Camb.~Univ.~Pr.~1993, \isbn{0\,521\,45839\,0}\\ {\em first part has thorough introduction to various flavors of incidence structure} \bitem{Cam94} \name{Peter J.\,Cameron}, \book{Combinatorics: topics, techniques, algorithms},\\ Camb.~Univ.~Pr.~1994, \isbn{0\,521\,45761\,0}\\ {\tt\PMlinkexternal{http://www.maths.qmul.ac.uk/~pjc/comb/}{http://www.maths.qmul.ac.uk/~pjc/comb/}} (solutions, errata \&c.)\\ {\em good combinatorics textbook, with detail} \bitem{Pot95} \name{Alexander Pott}, \book{Finite Geometry and Character Theory},\\ Lect.~Notes~in~Math.~\vol{1601}, Springer~1995, \isbn{3\,540\,59065\,X}\\ {\em includes clear introduction to incidence structures} \bitem{CD96} \name{Charles J.\,Colbourn} and \name{Jeffrey H.\,Dinitz}, eds.\\ \book{The CRC Handbook of Combinatorial Designs},\\ CRC~Press~1996, \isbn{0\,8493\,8948\,8}\\ {\tt\PMlinkexternal{http://www.emba.uvm.edu/~dinitz/hcd.html}{http://www.emba.uvm.edu/~dinitz/hcd.html}} (errata, new results)\\ {\em the reference work on designs incl.\ Steiner systems, proj.~planes} \end{thebibliography}