Fuji-Hara, Ryoh

Shinohara, Satoshi

Kyoto University Research Information Repository

TitleAlgebraic curves and balanced $n$-ary designs (AlgebraicCombinatorics)Author(s)Fuji-Hara, Ryoh; Shinohara, SatoshiCitation数理解析研究所講究録 (1999), 1109: 59-66Issue Date1999-08URL http://hdl.handle.net/2433/63310RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityAlgebraic Curves and Balanced $n$-ary Designs(Ryoh Fuji-Hara)(Satoshi Shinohara)AbstractLet $\mathcal{V}$ be a set of points and $B$ a collection of multi-subsets (calledblocks) of $\mathcal{V}$ each of size $k$ . A balanced $n$-ary design is a pair (V, $B$)such that each pointt occurs at most $n-1$ times in any block, andeach unordered pair of distinct points occurs $\lambda$ times in the blocks of$B$ . Note that in the block $\{x, x, y\}$ , the pair $\{x, y\}$ is counted twice inthe block. We show here some constructions of balanced $n$-ary designsby using algebraic curves over finite fields.1 IntroductionLet $\mathcal{V}$ be a set of $v$ points and $B$ a collection of multi-subsets, called blocks,of $\mathcal{V}$ . A balanced $n$-ary design is a pair (V, $B$) satisfying(1) each block is of a constant size $k$ ,(2) each point occurs at most $n-1$ times in any block $B\in B$ , and(3) each unordered pair of distinct points occurs exactly $\lambda$ times in the blocksof $B$ .Note that, for example, the block size of $B=\{x, x, x.y, y, z\}$ is 6 since thepoints $x,$ $y$ and $z$ occur 3 times, twice and once, respectively. And in the1109 1999 59-66 59block $B$ the pairs $\{x, y\},$ $\{y, z\}$ and $\{x, z\}$ are counted 6, 2 and 3 times,respectively.Let $N=(n_{ij})$ be a $|\mathcal{V}|\cross|B|$ matrix, where $n_{ij}$ is the number of occurrencesof the i-th point in the$.$$j$ -th block. We consider $N$ as the incidence matrix ofa balanced $n$-ary design. Using the incidence matrix, the conditions in thedefinition of a balanced $n$-ary design can be rewitten as follows:$(1^{}) \sum_{i}n_{ij}=k$ for any $j$ ,$(2^{})0\leq n_{ij}\leq n-1$ for any $i,j$ , and$(3^{}) \sum_{j}n_{ijj}n_{i}’=\lambda$ for any unordered pair $\{i, i’\},$ $i\neq i’$ .Example 1.1. Let $\mathcal{V}=\{a, b, c, d, e\}$ and $B$ be the collection of the followingblocks:$\{a, b, b, d, d, e, e, e\},$ $\{a, a, c, C, C, d, e, e\}$ ,$\{b, b, b, c, C, d, d, e\},$ $\{a, a, b, c, c, d, d, d\}$ ,$\{a, a, a, b, b, c, e, e\},$ $\{a, c, c, d, d, d, e, e\}$ ,$\{a, a, b, b, b, C, d, d\},$ $\{b, b, c, C, d, e, e, e\}$ ,$\{a, a, b, b, c, c, c, e\},$ $\{a, a, a, b, d, d, e, e\}$ .Then (V, $B$) is a balanced 4-ary (quaternary) design with 5 points, 10 blocks60and the block size is 8. The incidence matrix of the above balanced 4-arydesign is$[_{3}^{1}022$ $02231$ $30221$ $03221$ $20321$ $30221$ $02321$ $30221$ $03221$ $0]2231$ ,and for any pair $\{i, i’\},$ $i\neq i’$ ,$\sum_{j}n_{ijij}n’=23=\lambda$ .Let $\rho_{s}^{(i)}$ be the number of blocks containing the i-th point exactly $s$ times,i.e. the number of the entry $s$ in the i-th row of the incidence matrix. If$\rho_{s}^{(i)}=\rho_{s}$ for all $i$ then the design is said to be regular. If a balanced n-arydesign is regular then the replication number $R_{i}$ of the i-th point is a constantnumber $R$ , since$R_{i}= \sum_{f=0}^{-1}r\rho_{r}^{(i)}n=\sum_{\mathrm{f}}r\rho r=R$ .Balanced $n$-ary designs were first introduced by Tocher [3] in a statisticalpaper. The interested reader is referred to $[1, 2]$ for their excellent surveys.2 Algebraic curvesLet $q$ be a prime power, and $GF(q)$ a finite field of order $q$ . A divisor $D$ ona curve $C$ is a formal sum $\sum_{P\in c^{m_{P}}}P,$ $m_{p}\in Z$ . The set of divisors on $C$ is61denoted by $\mathrm{D}\mathrm{i}\mathrm{v}(D)$ . The support of a divisor $D$ , denoted by $\mathrm{S}\mathrm{u}\mathrm{p}\mathrm{p}(D)$ , is theset of points satisfying $m_{P}\neq 0$ . A divisor $D$ is said to be efficient if $m_{P}\geq 0$for all $P$ , and denoted by $D\geq 0$ . The degree of $D$ is $\deg D=\sum_{P}m_{P}$ . Let$D= \sum_{P}m_{P}P$ and $E= \sum_{P}m_{P}’P$ . Then $D+E= \sum_{P}(m_{P}+m_{P}’)P$ .Let Rat $(C)$ be the set of rational functions over a curve $C$ . The divisorof a rational function $f\in \mathrm{R}\mathrm{a}\mathrm{t}(C)$ is $\mathrm{d}\mathrm{i}\mathrm{v}(f)=\sum_{P\in C}mPP$ , where $m_{P}$ is theorder of $f$ at $P$ . For a divisor $D$ , a divisor $E$ is said to be equivalent to $D$ ,denoted by $E\sim D$ , if there exists a rational function $f\in \mathrm{R}\mathrm{a}\mathrm{t}(C)$ satisfying$E=D+\mathrm{d}\mathrm{i}_{\mathrm{V}}(f)$ .Let $C$ be a curve defined by $F=0$, where $F$ is a polynomial over afinite field $GF(q)$ . The divisor of the curve $C$ is the divisor of $F$ . Note that$\mathrm{d}\mathrm{i}\mathrm{v}(C)=\mathrm{d}\mathrm{i}\mathrm{v}(F)\geq 0$ . When the divisor of $C$ is written as $\mathrm{d}\mathrm{i}\mathrm{v}(C)=\sum_{p}m_{p}p$ ,the intersection multiplicity of a point $p$ on a curve $C’$ with the curve $C$ isthe order of $F$ at $p$ , say $m_{p}$ .Let $L(D)=\{f\in \mathrm{R}\mathrm{a}\mathrm{t}(C) : \mathrm{d}\mathrm{i}\mathrm{v}(f)+D\geq 0\}\mathrm{U}\{0\}$. It is well-known that$L(D)$ is a linear space with a finite dimension over an extension field $GF(q^{m})$$\mathrm{o}\mathrm{f}GF(q)$ .623 Algebraic curves and balanced $n$-ary de-signsLet $C$ be an irreducible curve defined over $GF(q)$ and $C=\{C_{1}, \cdots, C_{b}\}$ aset of $b$ curves defined over an extension $GF(q^{m})$ of $GF(q)$ . Let $V_{j}$ be a setof intersection points of $C_{j}\in C$ with $C$ , and $\mathcal{V}=\bigcup_{j}V_{j}=\{p_{1}, \cdots,p_{v}\}$ .Let $n_{ij}$ denotes the intersection multiplicity of a point $p_{i}$ with a curve $C_{j}$ .We consider the $v\cross b$ matrix $(n_{ij})$ as the incidence matrix. To satisfy theconditions of balanced $n$-ary designs, we have to choose suitable $C,$ $C$ and $\mathcal{V}$ .The first condition is required for the block size to be constant, i.e.,$\sum_{i}n_{ij}=k$ for any curve $C_{j}\in C$ . Let $D$ be a divisor on $C$ and $D=$$\{\mathrm{d}\mathrm{i}_{\mathrm{V}}(f)+D : f\in L(D)\backslash \{0\}\}=\{E_{1}, \cdot’. , E_{b}\}$ . Note that each element of$D$ is the divisor of a curve.Lemma 3.1. Let $C$ be the set of curves $\mathit{8}uch$ that their divisors are the el-ement8 of D. For any curve of $C$ , the total number of multiplicities of itsintersection points with $C$ is a constant number $k$ .Proof. The set $D$ is the set of efficient divisors being equivalent to $D$ , i.e.,$D=\{\mathrm{d}\mathrm{i}\mathrm{V}(f)+D:f\in L(D)\backslash \{0\}\}$$=\{E\in \mathrm{D}\mathrm{i}\mathrm{v}(C) : E\geq 0, E\sim D\}$.63It is well-known that if $E\sim D$ then $\deg D=\deg E$ . Hence for any $j$$\sum_{i}n_{ij}=\deg E_{j}=\deg D=k$ .$\square$We assume here that we choose $C$ as above, and that the point set $\mathcal{V}$of a design is the set $\mathcal{V}=\cup E\in v^{\mathrm{S}\mathrm{u}}\mathrm{p}\mathrm{P}(E)$ . The second condition says thateach point of the design occurs at most $n-1$ times. Since the intersec-tion multiplicities of points on curves are always positive, this condition isautomatically satisfied from Lemma 3.1.Theoren 3.2. Let $C$ be a curve defined over a finite field, $D$ a divisor on$C,$ $D=\{div(f)+D:f\in L(D)\backslash \{0\}\}=\{E_{1}, \cdot\cdot‘, E_{b}\},$ $\mathcal{V}=\bigcup_{E\in v^{S}}upp(E)$$E_{j}= \sum n_{ij}P_{i}$ , $P_{i}\in V$ . If $\deg C\leq 2$ and $D\geq 0$ then $(n_{ij})$ is the incidencematrix of a balanced n-ary design (V, $D$).Proof. We only have to check whether the third condition of balanced n-arydesigns is satisfied. Let $E_{j}$ be the j-th element of $D$ . Assume that $D$ and$E_{j}’ \mathrm{s}$ have the forms $D= \sum_{i}m_{i}P_{i}$ and $E_{j}=\mathrm{d}\mathrm{i}\mathrm{v}(f_{j})+D$ , respectively. Let$\mathrm{d}\mathrm{i}\mathrm{v}(f_{j})=\sum_{i}e_{ij}P_{i}$ . Then each entry $n_{ij}$ of the incidence matrix $(n_{ij})$ is$m_{i}+e_{ij)}$ since $\mathrm{d}\mathrm{i}\mathrm{v}(f_{j})+D=\sum_{i}e_{ij}P_{i}+D--\sum_{i}(m_{i}+e_{ij})P_{i}$ . For any pair64$\{i, i’\}$ , we have$= \sum_{j}’(m_{i}+e_{i}j)(mi’+e_{ij}’)\sum n_{ij}n_{i}jj$$= \sum_{j}(m_{i}m_{i’}+miei\prime j+m_{i}\prime eij+e_{i}je_{i’j})$$=bm_{i}m_{i^{J}}+m_{i} \sum_{j}e_{i}\prime j+mi’\sum je_{i}j+\sum_{j}eije_{i’}j$ .Let $\lambda(a, b)$ be the number of $j$ satisfying $(e_{ij’ j}e_{i}’)=(a, b)$ . Then we have$\sum_{j}e_{ijj}e_{i}’=\sum_{(a,b)}ab\lambda(a, b)$.When the degree of base curve $C$ is less than or equal to 2, it can be easilyseen that if $a+b=c+d$ then $\lambda(a, b)=\lambda(c, d)$ . Moreover we can see that$\sum_{j}e_{ij}’=\sum_{j}e_{ij}=\sum aa\lambda(a)$ ,where $\lambda(a)$ is the number of $j$ satisfying $n_{ij}=a$ . Since both of $\lambda(a, b)$ and$\lambda(a)$ are independent of $\{i, i’\}$ chosen, we have$\sum_{j}n_{ijj}n_{i}’=\lambda$,and we can conclude that the third condition in the definition of a balanced$n$-ary design is also satisfied. $\square$65References[1] E. J. Billington, Balanced n-ary designs: combinatorial survey and somenew $result_{\mathit{8}}$ , Ars Combin. 17A (1984), 37–72.[2] E. J. Billington, Designs with repeated elements in blocks: a survey andsome recent results, Congr. Numer. 68 (1989), 123–146.[3] K. D. Tocher, Design and analy8is of block experiments, J. R. Statist.Soc. B14 (1952), 45–100.66

Algebraic curves and balanced $n$-ary designs (Algebraic Combinatorics)

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