Following Eluer's method，A. Hedayat constructs some Knut
Vik designs. We call them Knut Vik designs of Hedayat in this
note. We give Knut Vik designs of Hedayat explicitly and decide
when Knut Vik designs of Hedayat are mutually orthogonal

Ishihara, Toru

Tokushima University Institutional Repository

J. Math. Univ. Tokushima Vo1.40 (2006)， 1-7 Construction of Knut Vik Designs and Orthogonal Knut Vik Designs By Toru ISHIHARA Fαculty olIntegr，αted Artsαnd Sciences， The Universitν01 Tokushimα， Minαmijosαnjimα， Tokushimα770・8502，JAPAN e・mαiladress: ishihαTα邸αs.tokushima-u.αc.jp (Received September 29， 2006) Abstract Following Eluer's rnethod， A. Hedayat constructs some Knut Vik designs. We call them Knut Vik designs of Hedayat in this note. We give Knut Vik designs of Hedayat explicitly and decide when Knut Vik d出ignsof Hedayat are mutually orthogonal. 2000 Mathematics Subject Classification. 05B15 Introduction Let A be a Latin square of order n， that is， an n x n array in which n distinct symbols are arranged 150 that each syrnbol occurs once in each row and colurnn. Index its rows and colurnns by 1，2，.. n. By the jth right diagonal of A we mean the following n cel of A: (i，j+i-1); i = 1，2γ ・，n; (rnod n.) We define also the jth left diagonal of A to the following n cel of A: (i，j-i); i = 1，2，.. ，n; (rnod n.) Let ~ be a set of n distinct syrnbols. If we can fil the cells of A by the elernents of ~ in such a way that each row， colurnn， right diagonal and left diagonal of A contains al the elernents of ~， we 15乱，ythe resulting structure a Knut Vic design， which we denote by K. It is also called a pandiagonal Latin sq凶 re[1]， [3]. In this paper， we set ~ = {O， 1，2，. . . ，n -1}. It is well known that Knut Vic designs of order何， exists if and only if n is not divisible by 2 or 3. A. Hedayat [3] showed that n is not divisible by 2 or 3 then K 早 (kij)with kij = >"i + j(rnod n) is a Knut Vic design if入，入-1，入+1 are relatively prirne 2 Toru Ishihara to n. In this paper， we call these Knut Vic designs as Knut Vic designs of Hedayat， Let n have the prime decompsition nzp?1P22..-pア.Then a1so he showed that there are N ニ pr-lpr-1.--pfv-1(Pl-3)(P2-3)・・・ (Prケ 3).different choices for入.In the present note， we define a standard way which gives入satisfingthe condition that入，入-1，入+1 are re1ative1y prime to n. K. Afsarinejad showed that there exist at 1east min(Pi -3)， (i = 1，2，...， r) mutually orthogona1 Knut Vik designs of order n. We show that there exist at most min(Pi -3)， (i = 1，2，"'， r) mutual1y orthogonal Knut Vik designs of Hedayat. We a1so obtain that to each Knut Vik design of Hedayat， there are pr-lpr-1 ・・・ p~r-1 (p1 -4)(p2 -4)・・・ (Pr-4) orthogonal Knut Vik de日ignsof Hedayat. 1. A standard construction of Knut Vik designs of Hedayat Let n be not divisib1e by 2 or 3 and have the prime decomposition η= pr1p~2 ・・・ p~r ， (3 < P1 < P2 <・・・ <Pr). If入，入十 1，入-1 are re1ative1y prime to n， then K = (kij) with kij = >'i + j (mod n) is a Knut Vik design of Heda抑 t.We decide explicit1y when入，入+1，入-1are relative1y prime to n. Put mzzら+1， 1壬α::;r， 1::;九三Pa-3. F)"om Chinese remainder theorem， we obtain Lemma 1. For 印 ch{i1，払・ .，ir}with 1三九三P1-3， 1 :;i2壬P2-3，・・ 11::;九三Pr-3， there isαpositive integer m satis，βes m = mt (mod pl)， m = m~2 (mod P2)，' • • ， m ニ mIr(mod Pr). Construction of Knut Vik Designs and Orthogonal Knut Vik Designs 3 As this integer is unique on Z(P1P2"'Pr) ，ωe denote it bνmili2'..ir・Now， we obtain Theorem 2 Set n1 = Pl1-1p22-1. .p~r-1. For each {t，i1，払・・" ir}切thO壬t:; n1 -1， 1 :;i1 :;P1 -3，1 :;i2 :; P2 -3，' . " 1:; ir三Pr-3， put 入(t，il， i2， • • • ，ir) = P1P2・.'Prt + mi1i2…ルthen theseλ's give N different choices for Knut Vik designs of Hedαyat. Following the proof of Chin邸 eremainder theorem， we construct explicitly integers mi1i2…ル蹴 follows.Put q1 = pi1 (mod P2)， q2= (P1P2)ー1(mod乃)"..，qr-1 =印1P2'"Pr-1)一1(mod Pr) Now we get inductively， mi1i2・ir = mt (mod Pl)， mi1i2・包r = m;l + P1S1 = m~2 (mod P2)， Sl = ql (m~2 -mU + S2P2， (_2 _1、mi1i2 = mi1 + P1 q1 (mi2 -mi1)， mi1i2・ir = mi山 +P1P282=m23(modP3)， 82=q2(mL-m叩 2)+ S3P3， 問 ilizia ニ mω2+PlP2q2(mL-mz142)，m九i2，.ir = m'i巾 ia+ P1P2P拘 =m乙(modP4)， mi1i2・ir_l = mi1i2・ir_2+ P1P2・・・PT-2qr-2(m;ニi-m4142・ir-2)'mi1i2・i，. = mi内 H ・ir-1+ P1P2・・・Pr-1Sr-1= mir (mod Pr)， Sr-l = Qr-l(mir -mi山・ ir-1)(mod (P1P2'" Pr))， mω2・ir = mi内ルー 1+ PIP2 ・・ Pr-lqr-l(m.~r -mi1i2・ir_1)mi1i2..ir_1(mod (PIP2' Pr)). Example 1. Let n = 5 x 7. Then Pl = 5，P2 = 7岨 dQ1 = 3 (mod 7). We have mi = 2 ， m~ = 3，mr = 2， m~ = 3 ， m~ = 4 ， m~ = 5. Hence， we get mi1i2 = m;l + 15(m;2 -m;J (mod 35)， ニ il+ 1 + 15(i2 -i1) (mod 35)， 1::; i1壬2，1三i2壬4.4 Toru Ishihara Thus， we obtain m11 = 2， m12 = 17， m13 = 32， m14 = 12， m21 = 23， m22 = 3， m23 = 18， m24 = 33 (mod 35). Example 2. Let n = 5 x 7 x 11.Then P1 = 5，P2 = 7，p:CJ = 11円ql = 3 (mod 7)， q2= 6 (mod 11). m{l' m~2 are the same出 inExample 1， and mLZ 43+l，1三i:CJ壬8.It is evIdent that mi1i2 1三九三 2，1壬i2壬4are also the same as in Example 1. Now we have mi1仇 =mh42+210(mL-mm)(mod385). Now we write simply mi山ホ for(丸山1，ii内 2，• • "令市8)'m11* = (2，212，37，247，72，282，107，317)， mゅ=(332，157，367，192，17，227，52，262)， m13* = (277，102，312，137，347，172，382，207)， m1仇=(222，47，257，82，292，117，327，152)， m2h = (233，58，268，93，303，128，338，163)， m22* = (178，3，213，38，248，73，283，108)， m2:h = (123，333，158，368，193，18，228，53)， m24率二 (68，278，103，313，138，348，173，383).Example 3. Let nニ 72X 1. Then Pl = 7，P2 = 11， ql= 8 (mod 11). mjlzi1+l，mL=ら+1， 1三らざ 4，1三i2:; 8. Hence， we get mi1i2 = m;l +56(mT2 -m}J = iI+1+56(i2-il) (mod 77)， 1三九三 4，1壬i2三8.We write mil* for (il1'九2γ.. ，i18)' mhニ (2，58，37，16，72，51，30，9)，m2* = (24，3，59，38，17，73，52，31)， m3* = (46，25，4，60，39，18，74，53)， m，仇=(68，47，26，5，61，40，19，75). Now we obtain 入(t，i1，i2)= 77t + mi1i2， 0三t三6，1壬九三4，1:; i2 :; 8. Construction of Knut Vik Designs and Orthogonal Knut Vik Designs 5 2. Orthogonal Knut Vik designs of Hedayat Let K1 and K2 be Knut Vik designs of Hedayat of order n. K1 and K2 are said to be orthogonα1 ifthey are orthogonal in the sense of Latin squares. Using t臨h恥en削O叫t刷aare given by kS)=λ(ωi~;1)，・，ti句 kg)=入(tμi2)，ti2)，，ti叩where 0壬t1壬η1-1，1壬ijl)三P1-3， 1壬ti1)壬P2-3，・，1三41)壬prー 3，0壬t2:s n1 -1，1壬42)壬P1-3， 1三42)5P2一色…，1三tp)壬Pr-3. It is known that K1 and K2 are orthogonal if and only if λ(tlJi勺~1) ，...， i~1)) 一入(t2 ， i~2) ， í~2) ， • .• ， i~2)) and n are relatively prime. Hence we h払veLemma 3. Knut Vik designs 01 Hedα仰 tK1αndK2αre orthogonalザandQnlyザtil)μi2)，41)チ42)，…，ti1)利子)When K1 and K2 are orthogonal ， we call (K1.K2) a orthogonal Pαirin this note. From Lemma 3， itfollows Theorem 4. For each Knut Vik design 01 Hedαyat， thereαre pr，-1p22-1. .p~r-1(pl -4)(p2 -4)・ (Pr-4) K nut Vik designs orthogonα1 toit. Thereαre jpialサ 2-2 P7白r-2(p1-3)印1-4)(p2一拍-4) 印T一州一4)orthogonal pairs 01 Knut Vi・kdesigns 01 Hedayat . Let S be a set of Knut Vik designs of order n. We say S to be a set of mutually orthogonal Knut Vik designs of order n if any two Knut Vik designs in S are orthogonal. It is shown by K. Afsarinejad that there are at least P1 -3 mutually orthogonal Knut Vik designs of order n. On the other hand， by Lemma 3， there are at most P1 -3 mutually orthogonal Knut Vik designs of Hedayat. In fact， P1-3 mutually orthogonal Knut Vik designs are given by K1 = (は¥K2=(的))， ，km-3=(砂川))， where 6 Toru Ishihara kS)=入(thlA1)，…，tiO)t+j， kg)=入(ω，ti2)，…，ti幻)i+ j，…， K314)=入(tm-3J1-3，r-3)，・，tir3))i+j，and 4a)チグ)，tr)引;b)，…，tia)チ4P)， for anyαチb.Thus， we obtain Theorem 5. The mωim仰 nnumber 01 mutuαly ortlwgonal K nut Vik designs 01 Hedαyαt isPl -3. There a問(prl-lp~~-l . . p~r-l)Pl-3 (P2-3)(p2-4) ・・・ (p2-pl+ 1)(Pa-3)仇-4)・(P3-Pl+1)-・ (Pr-3)(Pr -4)・ (Pr-Pl + 1)sets 01 Pl -3 mutuallνorthogonal Knut Vik designs 01 Hedayat. Example 4. Let n = 35. Then， the maximum possible number of mutually orthogonal Knllt Vik designs of Hedayat i白2.The al orthogonal pairs are given by the following pairs of入、-(入(1，1)= 2，入(2，2)= 3)， (入(1，1)= 2，入(2，3)= 18)， (入(1，1)= 2，入(2，4)= 33)， (入(1，2)= 17，入(2，1)= 23)， (入(1，2)=17，入(2，3)= 18)， (入(1，2)= 17，入(2，4)= 33)， (入(1，3)= 32，入(2，1)= 23)， (入(1，3)= 32，入(2，2)= 3)， (入(1，3)= 32，入(2，4)= 33)， (入(1，4)= 12，入(2，1)= 23)， (入(1，4)= 12，入(2，2)= 3)， (入(1，4)= 12，入(2，3)= 18). Let n = 5 x 7 x 1.Then， the maximllm possible number of mutually orthogonal Knut Vik designs of Hedayat is also 2. There are 672 orthogonal pairs in this case. the maximum possible number of mutually orthogonal Knut Vik designs of Hedayat is also 2. Let n = 72 X 11. IJ1 this case， the maximum possible number of mutually orthogonal Knut Vik designs of Hedayat is 4. There are 24 x 3 x 73 orthogonal pairs and 24 x 3 x 5 X 75 sets of 4 mutually orthogonal Knut Vik designs of Hed札，yat.Construction of Knut Vik Designs and Orthogonal Knut Vik Designs 7 References [1] K. Afsarinejad On Mutually orthogonal Knut Vik Designes， Statistics and Probゆi1ityLetters 5(1987)， 323・324.[ロ凶2司]J. De Theory and Applications， North-Holland， Amsterdam， 1991. [3] A. Hedayat， A Complete Solution to the Existence and Nonexistence of Knut Vik Designes and orthogonal Knut Vik D出 igns，J. Combinatorial The町y(A)22(1977)， 331・337.

Construction of Knut Vik Designs and Orthogonal Knut Vik Designs

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Construction of Knut Vik Designs and Orthogonal Knut Vik Designs

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