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bvpޡ@~PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!'Ռdrs/downrev.xmlPKP @B '  s *1#"@PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!'Ռdrs/downrev.xmlDAo0 HA!i4h* PI;~~}ygq!k QRj82)7L+&3TPfR"~d[x;Zg0ѕR;Fs$K5DŽ [z8FoWs'Ԡ-A¿; bvpޡ@~PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!'Ռdrs/downrev.xmlPKP  3333y___PPT10Y+D=' = @B +"  3+$(  $5 $  xV0e0e ?Rectangle 2<7I   iThe Existence of G(r,c)-Designs  6   $  x@w 0e0e ?Rectangle 3P  ,If a G(r,c)-design of order n exists then the following conditions hold: (a) rc n, (b) r+c-2 divides n 1, and (c) rc(r+c-2) divides n(n-1).tI N I((('((  3333y___PPT10Y+D=' = @B +">  ( ((  (> (  x0e0e ?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!W^̤drs/downrev.xmlDMK@Eax;;Xc%X -ByM7aflR ]^\b5Nֲ$A\YrYcgjyyLہwt*C-"} LJ_5dOlOuCᦓIJ-LJ{zl,MHnv]T*u}5 nW^~8?\w9/FSPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!W^̤drs/downrev.xmlPKUH$D .   \Lattice Rectangles 6  (  x`dk0e0e ?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDMO0DHkQ-6ԭ|K\mmב)G3zX ua+B#style.visibility<*(%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D' = @B%(D' = @B%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(]%(D' =+v4 ,0B*O3> B<*(]D' = @B%(D' = @B%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(]%(D' =+v4 ,0B*O3> B<*(]D' = @B%(D' = @B%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+v4 ,0B*O3> B<*(+p+0+( ++0+( +"    , (  ,5 ,  x0e0e ?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!}drs/downrev.xmlD[K1C7Z֦XJ^p3Yn >3|ol'CXx δ\+nf BD69&' \\^17[: !GM}.e Y #rbLx<&mMŖB==6o`,~u[O .V " zgW^]6N6ӷf!W.6\PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!}drs/downrev.xmlPKUH$ .   T Real World  6  J ,  x =H0e0e ?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!TT drs/downrev.xmlLN1E&CLI;!:Rn6LuN_Oジ7L؁|JF)m j,D ;gI‰g7S,;ڒذ@ m}y[2F'c1Ms!&ܠŞZ{#a j)b?g.,CµC}( cevkUb%$dL.PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!TT drs/downrev.xmlPK  H$.  jDIn most practical uses, the grid-block has size limitation and n is large. Thus, we have to consider the G(r,c) s with r < n1/2 and c < n1/2 , while preserving the unique collinearity condition, i.e. every pair of vertices occur at most once in the same row or column. We need a design!# K$G   (N#  3333___PPT10.+D' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D0' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' = @B%(D' = @B%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,K%(D' =+v4 ,0B*O3> B<*,KD' = @B%(D' = @B%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,K%(D' =+v4 ,0B*O3> B<*,KD' = @B%(D' = @B%(DG' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,#%(D' =+v4 ,0B*O3> B<*,#+p+0+, ++0+, +"    0 (  0@ 0  x@w 0e0e ?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Ndrs/downrev.xmlDQK0Co.lec8DCXpMbsSzx8-׃ę|h+N2ڙk vIH֫%]xO*"A8/ !azԝS4/ n;yerzhgeE,Y鶥Qۛa"n>j?+QFl1r÷f!Wm2PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Ndrs/downrev.xmlPK UH$D .   `Known Grid-Block Designs  ^ 0  x#0e0e ?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!drs/downrev.xmlDN1E&CLI˨# *aذ{L3i;i | Yܛss޴@>hg% )mk r0"Z$Dۛ)mAUYؐ&.w"[Է41i*0D_MgPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!drs/downrev.xmlPK  H$.  ~~A G(2,2)-design of order n exists if and only if n 1 (mod 8). This is also known as the 4-cycle system of order n. A G(2,3)-design of order n exists if and only if n 1 (mod 9). (J. E. Carter, 1989) A G(3,3)-design of order n exists if and only if n 1 or 9 (mod 36). 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(Mutoh, Morihara, Jimbo and Fu, SIAM Discrete Math., 2003) >I = 9=  3333___PPT10i.Gn+D=' = @B +"V  ]U<=(  <Z <  xED0e0e ?Rectangle 2<    n 1(mod 36) 0 0 0(   <  x` 0e0e ?Rectangle 3    The proof can be obtained right away if there exists an (n,9,1)-BIBD of order n = 36k+1. (See it?) The proof can also be obtained by using the existence of a (k,s,1)-BIBD, a G(3,3)-design of order 37 and a G(3,3)-design of Ks(36). (One vertex in common)  Z ZZ L    3333___PPT10i.0lk+D=' = @B +"  @(  @7 @  xED0e0e ?Rectangle 2   kCyclic Constructions    " @  xz 0e0e ?Rectangle 3   VA cyclic (72t+1,3,1)-BIBD and a cyclic (72t+37,3,1)-BIBD exist. Use Peltesohn s result (1938), we have five classes of base blocks for 72t+1 case: (a) for x = 0, 1, & , 3t-1, we have (0,1+2x,33t+1+x), (0,9t+1+2x,27t+1+x) and 0,9t+2+2x,18t+2+x), (b) for x = 0, 1, & , 3t-2, we have (0,2+2x,24t+2+x) and (c) (0,6t,24t+1). Use an imagination to put them together as grid-blocks (mainly by Mutoh).  Z MCWC(((C(  (8   3333___PPT10i.`+D=' = @B +"  D(  D  D  xz 0e0e ?Rectangle 2U   >  | D  x'0e0e ?Rectangle 3   PG(3,3)-Design Outline of proof. 1. A G(3,3) design of order 9 exists. 2. Let p be an odd prime and v p(mod 2p(p-1)). If there exists a cyclic (v,p,1)-BIBD, then there exists a G(p,p)-design of order pv. (F-H-J-M-S) 3. A cyclic (12k+3,3,1)-BIBD exists. A G(3,3)-design of order 36k+9 exists.   P P P$$$$C9N )< (   3333___PPT10i.[m+D=' = @B +")  @8 H (  H% H  xpv 0e0e ?Rectangle 2U   Y4x4 Grid-Blocks 0  H  xpv 0e0e ?Rectangle 3@`  fBy using the following grid-block we obtain a 4x4 grid-block design of order 97. (Difference Method) .g R0g ! 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P  xk0e0e ?Rectangle 3@  Theorem If a G(r,c)-design of Kn+1 and Km(n) exists respectively, then a G(r,c)-design of Kmn+1 exists. So, for G(4,4)-designs, we need a design of K97 to start with and we have constructed earlier. It s left to find a G(4,4)-design of Km(96) for proper m. Zg Z Z.63A#   P l??AutoShape 6PG  6   3333y___PPT10Y+D=' = @B +"I  `XPT@(  TB T  x`0e0e ?Rectangle 2U   vContinued & .  A   T  x@)0e0e ?Rectangle 3  Proposition For m 4, a G(4,4)-design of Km(96) exists. Note A 4-GDD of type 24m exists for each m 4. Using this fact and a G(4,4)-design of K4(4) we can prove the proposition. (By inflating a point into 4 points.) 2       ((((&  3333y___PPT10Y+D=' = @B +"O  f^`XF(  X( X  x`k0e0e ?Rectangle 2U   \With Two Missing Cases   X  x0e0e ?Rectangle 3   Theorem A G(4,4)-design of Kv exists if and only if v 1 (mod 96) except possibly v = 193 and v = 289. This is by the reason that the proposition works only for m 4. These two cases have been found recently by Zhang, Ge, Fu, Ling and Mutoh. Z Z((((2(:(7  3333y___PPT10Y+D=' = @B +"   p\(  \ \  xfL0e0e ?Rectangle 2U   Qv = 193 0 k \  xgL0e0e ?Rectangle 3P  -The key array we apply to find the solution. @. 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LJ_dЏlGuC%M+$H LJ;T~_F.pOa{m^Qۛ~=G+Q/ZtSm4PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!e-drs/downrev.xmlPKUH$ .   k4x4 Grid-Block Design     d  x x0e0e ?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!Wddrs/downrev.xmlDN1E&CLI!yAe9 wc:m'mq˛{snb5N,x`[;m3!U9Z,PlK>mc#ĆBPl(\6u }#sNY( iZ{{4{觬23_OfwC{9ݔE}(xO@/{UI!GH~6\^PK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!Wddrs/downrev.xmlPK  H$ .  +Combining the works above we have proved the following theorem. Theorem. A 4x4 grid-block design of order v exists if and only if v is congruent to 1 modulo 96.X ?(=(  3333___PPT10u.+tD' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D0' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dD' = @B%(D0' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d@%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*d@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*d@D' = @B%(D0' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d@%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*d@D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*d@+p+0+d ++0+d +".  E=h%(  h7 h  x0e0e ?Rectangle 2U   k3x4 Grid-Block Design    h  xx0e0e ?Rectangle 3   If a 3x4 grid-block design of order v exists, then v 1, 16, 21, 36 (mod 60). This is a joint work with Zhang, Ge, Ling and Mutoh. v = 16 is not possible. (?) For v 1, 21 (mod 60), it is settled. For v 16, 36 (mod 60), we only have 12 finite cases left unsettled.  Z55(< (%(@   3333y___PPT10Y+D=' = @B +"P  WOl7(  l# l  xl0e0e ?Rectangle 2Z*   W  $   l  x`l0e0e ?Rectangle 3   vA grid-block packing of order n is said to resolvable if the collection of grid-blocks can be partitioned into subclasses R1, R2, & , Rt such that every vertex of Kn is contained in precisely one grid-block of each class. Each Ri is called a resolution class. Clearly, such a packing exists only when rc divides n. :< + E  C  ?  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W Group Testing 6  |  x@i0e0e ?Rectangle 3"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!#=drs/downrev.xmlDN0DHuQ"V+B#style.visibility<*|%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*|D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*|D"' = @B%(D' = @B%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|I%(D' =-6B+checkerboard(across)*<3<*|ID"' = @B%(D' = @B%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|If%(D' =-6B+checkerboard(across)*<3<*|IfD"' = @B%(D' = @B%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|f>%(D' =-6B+checkerboard(across)*<3<*|f>D"' = @B%(D' = @B%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*|>P%(D' =-6B+checkerboard(across)*<3<*|>P+p+0+| ++0+| +"2"  9 1  (  <   x/0e0e ?Rectangle 2"PK![Content_Types].xml|N0 HC+jS8 @:Q۸ ??\o<7p]V޷O(I'6E=, ){Ō4>I)q7UuDހǏ)}nL"Mᴸ&g0eSA)΄2';2 WYu7{ɯΒzŘqFrj9K*(]mOlPK!Z,[ _rels/.relslj0 ``t_Pƈ[>,dgzjǎ?I'f#®Pb-\/Ƿ0Z]nLnp__3.iJV KQBiDžrL,Vʌ/7р4`ANar+m;E/'3U Aںv83/PK!+drs/downrev.xmlDN0HHܨCU*5,6בmڔь7[rz QP힮AL Na1??a푷t(C#"} \J_dO@uC1M/$ɤCZ?/`,a9ݭo .  yUW.ISӇ} Eh rPK-![Content_Types].xmlPK-!Z,[ _rels/.relsPK-!+drs/downrev.xmlPKUH$D .   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(r + c tests for a plate)T5  ~(5  3333___PPT10i.+%wD' = @B D' = @BA?%,( < +O%,( < +D' = @B%(D0' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' = @B%(D*' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*D' = @B%(D*' = @B%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*5D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*5D' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*5D' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*5+p+0+ ++0+ +"j  y..a(  )   xd0e0e ?Rectangle 2U   ]Basic Matrix Method 0    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Grid-Block Designs and PackingsGraph DecompositionWell-Known H-Designs K4-Designs Grid-Block An Example The Existence of G(r,c)-DesignsLattice Rectangles Real WorldKnown Grid-Block DesignsA Cyclic G(2,4)-Design2x4 Grid-Block Designsn k 1(mod 36) Cyclic Constructions PowerPoint ²4x4 Grid-BlocksA G(4,4)-Design of K4(4)!n+1 mn+1 Construction Continued KWith Two Missing Casesv = 193 v = 2894x4 Grid-Block Design3x4 Grid-Block Design A G(4,4)-Packing of order 16 PowerPoint ²Ready for Tests?Group TestingLibrary ScreeningBasic Matrix MethodMore Positive ClonesUnique Collinearity ConditionKeep Moving! 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