On a Grouping Method for Constructing Mixed Orthogonal Arrays

Transcription

1 Ope Joural of Saiic hp://dxdoiorg/1046/oj010 Publihed Olie April 01 (hp://wwwscirporg/joural/oj) O a Groupig Mehod for Corucig Mixed Orhogoal Array Chug-Yi Sue Depare of Maheaic Clevelad Sae Uiveriy Clevelad USA Eail: cue@cuohioedu Received Jauary 0 01; revied February 19 01; acceped March 8 01 ABSTRACT Mixed orhogoal array of regh wo ad ize are coruced by groupig poi i he fiie projecive geoery PG 1 PG 1 ca be pariioed io fla i aociaed wih a poi i PG 1 -fla uch ha each A orhogoal array 1 1 uig 1 1 poi i PG 1 fla over GF() if i i ioorphic o PG 1 If here exi a A e of 1 1 poi i ca be coruced by PG i called a 1 - -fla over GF() i PG 1 he we ca replace he correpodig by 1 1 -level colu ad obai a ixed orhogoal array May ew ixed orhogoal array ca be obaied by hi procedure I hi paper we udy ehod for fidig dijoi 1 -fla over GF() i PG 1 i order o co- ruc ore ixed orhogoal array of regh wo I paricular if ad are relaively prie he we ca co- 1 ruc a i 1 i for ay 0 i New orhogoal array of ize ad 104 are obaied by uig PG(7 ) PG(8 ) ad PG(9 ) repecively -level colu i 1 1 Keyword: Fiie Field; Fiie Projecive Geoery; 1 -Fla over GF() i PG 1 ; Geoeric Orhogoal Array; Marix Repreeaio; Miial Polyoial; Orhogoal Mai-Effec Pla; Priiive Elee; Tigh 1 Iroducio Orhogoal array of regh wo are ued a orhogoal ai-effec pla i fracioal facorial experie I a orhogoal ai-effec pla he ai effec of each facor ca be opially eiaed auig he ieracio of all facor are egligible e N 1 k deoe a orhogoal array of regh wo wih N row k colu ad i level i he ih colu for i 1 k I every N ubarray of N 1 k all poible cobiaio of level occur equally ofe a row I i kow ha N 1 i 1 i a N 1k ad he orhogoal array i called igh if he equaliy hold Orhogoal array N 1 k i called yeric if 1 oherwie i i called k ayeric or ixed Syeric orhogoal array have bee coruced i [1-] Mixed orhogoal array were iroduced i [4] ad hey have draw he aeio of ay reearcher i rece year Mehod for corucig ixed orhogoal array of regh wo have bee developed i [5-9] ad ay oher auhor Thee ehod ue Hadaard arice differece chee geeralized Kroecker u fiie projecive geoerie ad orhogoal projecio arice We refer o [10] for ore corucio ad applicaio of orhogoal array The ehod of groupig wa ued i [11] o replace hree wo-level colu i yeric orhogoal array by oe four-level colu for corucig ixed orhogoal array havig wo-level ad four-level colu A yeaic ehod [1] wa developed for ideifyig Copyrigh 01 SciRe

2 C-Y SUEN 189 dijoi e of hree wo-level colu for corucig N ( 4 ) The ehod wa geeralized i [6] for co- rucig 1 k r1 r where i a prie power Mixed orhogoal array of regh were coruced by uig ixed pread of regh i fiie geoerie i [1] Thi ehod wa alo idepedely dicovered i [14] for corucig ixed orhogoal array of regh hree ad four Orhogoal array coruced by hi ehod are called geoeric Geoeric orhogoal array 64 ( ) 64 ( ) 64 ( ) ad 64 ( ) were coruced i [1] However he ehod i rericed o corucig ixed orhogoal array wih he uber of level i each colu a power of I hi paper we hall ue fiie projecive geoerie o develop a geeral procedure for corucig ore ixed orhogoal array Moreover he procedure allow u o coruc ixed orhogoal array wih he uber of level i each colu a power of ay give prie uber We ar wih a yeric orhogoal array 1 1 ad he coruc ixed orhogoal array by replacig a group of colu wih aoher group of colu Our groupig ehod ue properie of fiie projecive geoerie which i differe fro he groupig ehod i [6] Hece we are able o obai oe ew erie of ixed orhogoal array Geoeric Orhogoal Array For r 1 ad a prie power le PG r 1 deoe he r 1-dieioal fiie projecive geoery over he Galoi field GF() A poi i PG r 1 i deoed by a r-uple x 1 xr where x i are elee of GF() ad a lea oe x i i o 0 Two r-uple repree he ae poi i PG r 1 if oe i a uliple r of he oher Hece here are 1 1 poi i PG r 1 A 1 -fla i PG r 1 i a e of 1 1 poi which are liear cobiaio of idepede poi A pread F of 1-fla of PG r 1 i a e of 1-fla which pariio PG r 1 I i kow [15] ha here exi a pread F of 1-fla of PG r 1 if ad oly if divide r We call a e of fla F1 Fk a ixed pread of PG r 1 if i pariio PG r 1 ad a lea wo fla i F have differe dieio Mixed pread are ueful for corucig ixed orhogoal array of regh wo Specifically we give he followig heore for corucig a orhogoal array fro a (ixed) pread The heore i he pecial cae of regh wo of Theore 1 [14] i fiie projecive geoery laguage Theore 1 e F 1 Fk of PG r 1 be a (ixed) pread where F i i a i 1-fla for i 1 k The we ca coruc a orhogoal array 1 k r We ow decribe he procedure o coruc he orhogoal array i Theore 1 For i 1 k le G i be a r i arix uch ha he i colu are ay choice of i idepede poi of he i 1-fla F i e G be he r i arix k G1 Gk The r 1 coi of r row which are he elee of he row pace of G where he i -level colu of G i i replaced by a i -level colu for each i 1 k We call orhogoal array geoeric if hey ca be obaied by Theore 1 Geoeric orhogoal array have bee coruced i [1 9116] Exaple of geoeric orhogoal array are: r 1 1 1) r ; ) r 1 1 r if divide r; r 1 r ) r if r ; ad k l 4) r where 1 1 j l 1 1 k j i j 1 r i j 0 j Mai Reul I i proved i ea 1 [1] ha if V 1 V V are hree dijoi 1 -fla of PG 1 he heir uio ca be regrouped io 1 dijoi 1-fla Hece hree 1 -level colu i a ca be replaced by 1 4-level colu By applyig hi reul o a pread of -fla of PG(5 ) 64 ( ) ad 64 ( ) were coruced Geeralizig he idea we would like o fid a ufficie codiio ha a e of fla i PG 1 ca be regrouped io a e of fla Sice here exi a pread of 1-fla of PG 1 we ca by Theore 1 coruc a If here exi 1 -fla i he pread uch ha heir uio ca be 1 1 -fla he we 1 regrouped io ca replace he correpodig 1 1 -level Copyrigh 01 SciRe

3 190 C-Y SUEN colu i he by -level colu ad obai a By repeaig hi proce ay orhogoal array ca be obaied Fir we would like o eablih a oe-o-oe corre podece bewee he 1 1 dijoi 1 - fla i PG 1 ad he 1 1 poi i PG 1 e ω be a priiive elee of GF( ) ad le he iiu polyoial of GF be where 0 1 are elee of GF() The copaio arix of he iiu polyoial i a arix W If ω i a priiive elee of GF he 01 are he elee of GF The elee of GF ca be repreeed by arice wih erie fro GF() The elee ω i i repreeed by W i ad he elee 0 ad 1 are repreeed by he zero arix ad he ideiy arix repecively Deoe he arix repreeaio of a elee x i GF by W(x) e each poi x 1 x PG 1 correpod o he 1 PG 1 -fla i which coi of poi ha are liear cobiaio of row vecor of he arix W x1 W x over GF() I ca be how ha he fla correpodig o he 1 1 poi of PG 1 pariio PG 1 Thi eablihe a oe-o-oe corre podece bewee he 1 1 dijoi 1-fla i PG 1 ad he 1 1 poi i PG 1 Defiiio 1 A e of 1 1 poi i PG 1 i aid o be a 1 -fla over GF() if i i poible o fid coordiae for hi e of 1 1 poi uch ha i i ioorphic o PG 1 over GF() i Noe ha wheher a e of 1 1 poi i PG 1 i ioorphic o PG 1 over GF() deped o oly o he choice of he poi bu alo o he choice of he coordiae for hee poi For exa- ple he e S1 1 1 i Exaple 1 (give afer Theore ) i a 1-fla over GF() i PG 1 8 ice i i ioorphic o PG 1 over GF() Bu if we chooe differe coordiae for 4 S he i i o ioorphic o PG 1 over GF() Hece i i ipora o pecify he correc coordiae whe a 1 -fla over GF() i PG 1 i eioed Alo we oe ha i i poible o have > for a 1 -fla over GF() i PG 1 For exaple S 1 ad S i Exaple (give afer Theore ) are -fla over GF() i PG 1 16 We ow give a ufficie codiio ha a e of 1 1 dijoi 1 -fla i PG 1 ca be regrouped io a e of 1 1 dijoi 1 -fla Theore A e of 1 1 dijoi 1 - fla i PG 1 ca be regrouped io a e of fla if he e of dijoi 1 1 correpodig poi i PG 1 i a 1 -fla over GF() Proof e he coordiae of he 1 1 correpodig poi of he 1 -fla over GF() i PG 1 x x for be 1j j j 1 Alo le be a arix uch ha he row are he poi of PG 1 The he 1 -fla i PG 1 correpodig o he poi x 1 j xj coi of poi which are he row of he 1 1 arix M j W x1 j W x j where W(x) i he arix repreeaio of x We ca verify ha for each i poi he e of which coi of he ih row of M M 1 i a fla i PG(-1 ) Noe ha i geeral here are ore way of regroupig 1 1 dijoi 1-fla i a e of Copyrigh 01 SciRe

4 C-Y SUEN 191 PG 1 io dijoi fla if he 1 1 i a 1-fla over GF() e P ij be he poi i PG 1 wih he ih row of M j a i coordiae The array of poi P P ij ha he followig properie: 1) Each row (colu) of P i a 1-fla ( 1 - fla) u 1 1 poi i a give row (colu) correpodig poi i PG 1 ) If for a 1 u u -fla he he 1 1 poi a he ae poiio i ay oher row (colu) alo for a u 1-fla For exaple if here exi a -fla over GF() i PG(1 16) he each of he 7 poi i he -fla over GF() correpod o 15 poi i PG(7 ) The 105 poi i PG(7 ) correpodig o he -fla over GF() i PG(1 16) ca be arraged io a 15 7 array uch ha each row i a -fla ad each colu i a -fla Sice a -fla ca be pariioed io five 1-fla he 15 7 array of poi ca be pariioed io five 7 ubarray uch ha each colu i a 1-fla ad each row i a -fla Alo coider a 15 ubarray of he 15 7 array uch ha each row i a 1-fla We ca elec a 7 ubarray uch ha each colu i a -fla Each of he reaiig eigh row i a 1-fla Hece he 15 ubarray ca be pariioed io hree -fla ad eigh 1-fla Therefore hee 105 poi ca be grouped io: 1) 7 i -fla ad 5i 1-fla for i 0 7 ; ) 15 i -fla ad 7i 1-fla for i 0 5 ; or ) four -fla hree -fla ad eigh 1-fla 6 Exaple 1 e 01 be he 8 elee of GF(8) wih 1 Coider PG(1 8) wih ie poi (0 1) (1 0) (1 1) (1 ω) (1 ω ) (1 ω ) (1 ω 4 ) (1 ω 5 ) ad (1 ω 6 ) Each poi of PG(1 8) correpod o a -fla i PG(5 ) ad he ie -fla pariio PG(5 ) We ca coruc a 64 (8 9 ) by Theore 1 e S1 1 1 S 1 1 S ad We ca verify ha S 1 S ad S are dijoi 1-fla over GF() i PG(1 8) The arix repreeaio W of ω ad he 7 arix give i he proof of Theore are W ; The hree poi of S 1 correpod o he hree -fla i PG(5 ) which are row of he followig hree arice M 1 M ad M repecively M1 W 1 W M W W M W W We oberve ha for each i 1 7 he ih row of M 1 M ad M are hree poi of a 1-fla i PG(5 ) Hece we ca replace he hree 8-level colu correpodig o S 1 i 64 (8 9 ) by eve 4-level colu o obai a ( ) Coiuig hi procedure we ca replace he hree 8-level colu correpodig o S i 64 ( ) by eve 4-level colu o obai a 64 ( ) Noe ha 64 ( ) ad 64 ( ) were alo coruc i [1] uig a differe ehod However Theore i ore veraile a how i followig exaple 14 Exaple e 01 be he 16 elee of GF(16) wih 4 1 Coider PG(1 16) wih 17 poi (0 1) (1 0) (1 1) (1 ω) (1 ω 14 ) Each poi of PG(1 16) correpod o a -fla i PG(7 ) ad he eveee -fla pariio PG(7 ) We ca coruc a 56 (16 17 ) by Theore 1 e S1 1 1 S 1 T Copyrigh 01 SciRe

5 19 C-Y SUEN ad T 1 1 xw i xw j xw i W j xw l T T 1 1 T 1 1 T We ca verify ha S 1 ad S are dijoi -fla ad T1 T5 are dijoi 1-fla over GF() i PG(1 16) Moreover S 1 S ad T 1 pariio PG(1 16) By he difollowig Theore we ca replace he ubar- cuio ray 56 (16 7 ) correpodig o S 1 or S i 56 (16 17 ) by 56 ( ) or 56 (8 15-i 4 7i ) 0 i 5 Siilarly we ca replace he ubarray 56 (16 ) correpodig o T1 T5 i 56 (16 17 ) by 56 (8 4 8 ) May ixed orhogoal array uch a 56 ( ) 56 ( ) 56 ( ) 56 ( ) 56 ( ) 56 ( ) 56 ( ) ca be obaied by hi procedure 4 Corucio of More Orhogoal Array I hi ecio ehod for fidig dijoi fla over GF() i PG 1 are developed o coruc ore orhogoa l array e α be a priiive elee of GF( ) ad le he arix repreeaio of α i GF() be W Sice ad i a elee of GF() W i he arix repreeaio of we have W I where I i he ideiy arix The for ay fixed poi x x 1 x i PG 1 he e i S xw : i 0 co i a o 1 1 poi x a 1 1 i PG 1 ice ( x I x) ad x repree he ae poi Moreover if β ad γ are ay elee of GF() ad xw i ad xw j are elee of S x he xw for oe l ice i j i βw + γw i he arix repreeaio of he elee βα + γα j of GF S x ha he rucure of a fla over GF() i 1 PG ice liear cobiaio of poi i S x are alo poi i S x I fac S x i a 1-fla over GF PG 1 if ad oly if he uber of () i poi i S x i 1 1 x ad y are wo poi i PG 1 for oe ieger Now if ad Sx Sy he here exi i ad j uch ha xw i = yw j i j We have y xw S x hece S x = Sy PG 1 ad le Theore e x be a poi i i x xw : i 0 1 i PG 1 uber of S x i 1 1 ay wo poi x ad y i PG 1 or Sx Sy Hece PG 1 S The S x i a -fla over GF() if ad oly if he poi i for oe ieger Moreover for eiher S x = S y ca be pariioed io dijoi e of S x Exaple We illurae how we obai he hree di- joi 1-fla over GF() i PG(1 8) i Exaple 1 e ω be a priiive elee of GF(8) wih ω = ω + 1 ad le α be a priiive elee of GF(4) wih α = α + 1 ad arix repreeaio The 0 1 W 1 1 S W 01W S 1 1 W 1 1 W 1 1 ad S 1 1 W 1 W S GF(4) 8 8 S S ad S ve dijoi 1-fla over GF() are T Exaple Theore 4 If i a prie power a 8 8 relaively prie he we ca coruc ixe 1 1 are hree dijoi 1-fla over GF() i PG(1 8) Exaple 4 e ω be a priiive elee of GF(16) wih ω 4 = ω + 1 ad le α be a priiive elee of wih α = α + 1 ad arix repreeaio W give i Exaple The 17 poi of PG(1 16) ca be pariioed io he followig fla over GF(): S S S 1 1 The fi T i array 1 5 d ad are d orhogoal Copyrigh 01 SciRe

6 C-Y SUEN 19 1 i 1 1 i for i Proof We ca coruc a 1 1 fro PG 1 Fro he proof of Theore 46 [15] if ad are relaively prie he S x i a 1 -fla over GF() i PG 1 for PG 1 PG 1 riioed io poi x i every Hece ca be pa 1-fla over GF() Each S x repree 1 1 -level colu i 1 1 ad by Theore i ca be re- placed by 1 1 -level colu The followig reul which follow fro Theore 4 i a geeralizaio of Theore 4 Corollary 1 If i a prie power ad d i he greae coo divior of ieger ad he we ca coruc ixed orhogoal array d d 1 1 d i 1 1 i 1 1 for 0 d d i Proof If d i he greae coo divior of ad he d ad d are relaively prie By ubiud d ad d repecively i ig ad wih Theore 4 we obai he ixed orhogoal array By uig Theore 4 ad Corollary 1 we obai he followig ew erie of igh orhogoal array for ay prie power 1) 6 ) 10 1 i 1 0 i + 1; 10 1 i ) i i i 1 ; 1 i i i 1 ; 4) i i 4 + 1; 14 1 i ) 6) i i i 1 i i 1 1 ; ad i 1 1 The followig heore give a e of 1 dijoi 1 -fla over GF() i PG(1 ) Theore 5 For i 0 le Ti i : GF \ 0 ive elee of GF The T T 0 are 1 di- joi 1 -fla over GF() i PG (1 ) Proof Ti i a e of 1 1 poi i PG(1 i i ( ) ice where ω i a prii- ) repree he ae poi for each ozero elee α of GF() To how ha T i i a 1 -fla over GF( ) we prove ha ay liear cobiaio of elee i Ti i agai i T i If i i GF he 1 1 T i ad 1 i i i T i i For 0 i < j if 1 1 T ad i j T j repree he ae poi i PG(1 ) i 1 j 1 he 1 j i 1 Hece 1 Bu 1 k 1 1 for oe 0 k which coradic 0 i < j Hece T i ad T j are dijoi for all 0 i < j 1 Corollary 1 i 1 1 i ca be coruced for ay ieger prie power ad i Proof We ca coruc a fr o PG(1 ) For each i 0 le Ti Ti be a - fla over GF() i PG(1 ) T i : i 0 i a e of 1 dijoi (-)-fla over GF() i PG(1 ) The for each T i we replace he correpodig Copyrigh 01 SciRe

7 194 C-Y SUEN level colu i 1 by 1 1 Exaple 5 e ω be he priiive elee of GF(16) wih ω 4 = ω level colu o obai he orhogoal array GF T : 16 \ i a -fla over GF() i PG(1 16) ad T 11 0 T i a -fla over GF() i PG(1 16) Noe ha we ar -fla ove G F() i PG( 1 16) i Exaple by rial ad error However we do o have a ehod o fid ore ha -1 dijoi (-)-fla over GF() i PG(1 ) Wih = ad 7 i Corollary we obai he followig ew erie of igh orhogoal array for ay prie power ad i ) i 1 1 i 1 8 ; e able o fid wo dijoi r ) 10 Theore 6 For ay ieger a i i 5 1 ; ) i i 6 1 ; ad 4) i i The followig heore give a -fla over GF() i PG( ) The proof i oied ice i i iilar o ha of Theore 5 d GF \ 0 : GF GF 00 T i a -fla over GF() i PG( ) fid ore dijoi -fla over GF() i PG( ) However for β 1 β he -fla over GF() T ad Theore 7 e ω be a priiive elee of GF( ) 1 T are o dijoi i PG( ) Bu if = we ca ad le : F : GF F 00 : 00 S GF GF T GF G 00 U G ad V GF GF The we have 1) S ad T are dijoi -fla over GF() i PG( ) for ) T U ad V are hree dijoi -fla over GF() i PG( ) if i eve Proof By Theore 6 S T U ad V are -fla over GF() i PG( ) We ow prove ha S ad T are di- joi Aue ha S ad repree he ae poi i PG( T ) where 1 GF ad 1 GF Clearly α 1 α γ 1 γ 0 hece α 1 = α = 1 ad 1 1 ad 1 repree he ae poi We have 1 ad 1 1 which iply ω = 1 a coradicio Hece S ad T are dijoi Now we how ha T ad U are dijoi if i eve If i eve he divide 1 For ay GF \ 0 0 k 1 1 Aue ha U where GF ad GF γ 0 hece α = 1 ad 1 he a γ = ω k for oe T ad repree he a e poi i PG( ) α γ α = ad Clearly α repree e poi We have γ = γ k ad which i ply for oe 0 k 1 1 a coradicio Hece T ad U are dijoi We ca i ilarly prove ha T ad V are dijoi ad ha U ad V are dijoi if i eve 1 A oruced fr ca be c o PG( ) By applyig Theore 6 ad 7 we obai he fol- Copyrigh 01 SciRe

8 C-Y SUEN 195 lowig orhogoal array Corollary For ay prie power we ca coruc ) 1 ; ) 1 1 ; ad ) Exaple 6 e ω be he priiive elee of GF(8) wih ω = ω + 1 e S T be wo dijoi -fla over GF() i PG(8) A 51 (8 7 ) ca be coru ced fro PG(8) We ca replace he ubarray 51(8 15 ) correpodig o S or T by a 51(16 7 ) o obai 51 ( ) ad 51 ( ) The followig wo exaple are obaied by applyig Theore ad 5 ad by rial ad error Exaple 7 e ω be he priiive elee of GF(8) wih ω = ω + 1 e ad A B C ad W be he arix defied i E xaple 1 For i 7 le A i (or B i C i ) be he e obaied by ulielee i A i (or B i C i ) by W For plyig each Exaple 6 A 1 W 01 W 1 0 W 1 01 I ca be verified ha A1 A7 B1 B7 C1 C are dijoi 1-fla over ad GF() i PG( 8) A 51 (8 7 ) ca be coruced fro PG( 8) We ca replace he ubarray 51 (8 ) correpodig o each 1-fla over GF() i PG( 8) by a 51(4 7 ) o obai 51 (8 7-i 4 7i ) for i 1 Exaple 8 e ω be he priiive elee of GF() wih ω 5 = ω + 1 A 104 ( ) ca be coruced fro PG(1 ) 1) A A 1 1 A 1 A A5 1 1 A6 1 1 A7 1 1 A 8 A9 1 1 A A ad are eleve dijoi 1-fla over GF() i PG(1 ) We ca replace he ubarray 104 ( ) correpodig o each 1-fla over GF() i he 104 ( ) by a 104 ( ) o obai 104 ( -i 16 i 4 16i ) for i 1 11 ) B1 1 B 1 B 1 1 B4 1 ad are four dijoi -fla over GF() i P G(1) We ca replace he ubarray 104 ( 7 ) correpodig o each - fla over GF() i he 104 ( ) by a 104 ( ) or a 104 (8 1 ) o obai 104 ( -7i-7j 16 7i 8 16i+1j ) for 1 i + j 4 Copyrigh 01 SciRe

9 196 C-Y SUEN ) 5 Dicuio C1 B1 C B are wo dijoi -fla over GF() i PG(1) where B 1 ad B are -fla over GF() i ) Moreover C 1 C ad A 1 i 1) pariio PG(1 ) We ca replace he ubarray 104 ( 15 ) correpodig o C 1 or C i he 104( ) by a 104 ( ) or a 104 (16 1 ) o obai 104 ( ) 104 ( ) 104 ( ) 104 ( 16 6 ) ad 104 ( ) We ue -fla over GF() i PG 1 o fid differe way o regroup a e of 1-fla i PG 1 io dijoi fla However ay proble reai uolved For exaple we do o kow how ay dijoi -fla over GF() exi i PG(1 ) Sice here are ( 1 1 1) poi i PG(1 ) he upper boud for he uber of dijoi -fla over GF() equal - if 4 ad equal + 1 if = A obviou cojecure i ha PG(1 ) ca be pari- ioed io -fla ad oe 1-fla over GF() Thi cojecure i rue for = ice PG(1 ) ca be pariioed io 1 1-fla ove r GF() by Theore 4 I i alo rue for = ad = 4 5 which are how i Exaple for = 4 ad how i Exaple 8() for = 5 If he cojecure i rue we ca coruc a 1 1 i 1 1 i for ad i 1 which would be a igifica iprovee of Corollary Alo we do o kow how ay dijoi -fla over GF() exi i PG( ) The uber of poi i PG( ) i Hece a upper boud for he uber of dijoi -fla over GF() i PG( ) i 1 if ad i + 1 if = The upper boud i aaied for = ice PG( ) ca be pariioed io 1 -fla over GF() by Theore 4 I geeral he differece be- boud ad wha ca be obaied i wee he upper Theore 6 ad 7 i coiderably large for There ay be beer way o fid dijoi -fla over GF() i PG( ) ha he approach ued i Theore 6 ad 7 So far we do o kow ay exaple havig 1 dijoi -fla over GF() i PG( ) for Aoher proble which cao be olved by he ap- ad proach of hi paper i he corucio of orhogoal array havig row wher e i a prie uber For exaple i i kow ha ( ) ca be coruced by a ixed pread of PG(6 ) which coi of a -fla ad 16 -fla Bu i i o kow ha if i i poible o fid aog hoe 16 -fla dijoi e of hree -fla uch ha each e ca be regrouped io eve 1-fla We could coruc a 18 (16 8 i 4 7i ) if here exi i uch dijoi e of hree -fla REFERENCES [1] R C Boe ad K A Buh Orhogoal Array of Sregh Two ad Three The Aal of Maheaical Saiic Vol No pp doi:10114/ao/ [] R Placke ad J P Bura The Deig of Opiu Mulifacorial Experie Bioerika Vol No pp 05-5 [] C R Rao Facorial Experie Derivable fro Cobiaorial Arragee of Array Joural of Royal Saiical Sociey (Supplee) Vol 9 No pp doi:1007/98576 [4] C R Rao Soe Cobiaorial Proble of Array ad Applicaio o Deig of Experie I: J N Srivaava Ed A Survey of Cobiaorial Theory Norh- Hollad Aerda 197 pp [5] J C Wag ad C F J Wu A Approach o he Corucio of Ayerical Orhogoal Array Joural of Aerica Saiical Aociaio Vol 86 No pp doi:1007/9059 [6] C F J Wu R C Zhag ad R Wag Corucio of Ayerical Orhogoal Array of Type OA( k ( r1 ) 1 ( r ) ) Saiica Siica Vol No pp 0-19 [7] A Dey ad C K Midha Corucio of Soe Ayerical Orhogoal Array Saiic & Probabiliy eer Vol 8 No 1996 pp doi:101016/ (95)0016- [8] Y S Zhag Y Q u ad S Q Pag Orhogoal Ar- of ray Obaied by Orhogoal Decopoiio of Projecio Marice Saiica Siica Vol 9 No 1999 pp [9] C Sue ad W F Kuhfeld O he Corucio Mixed Orhogoal Array of Sregh Two Joural of Saiical Plaig ad Iferece Vol 1 No 005 pp doi:101016/jjpi [10] A S Hedaya N J A Sloae ad J Sufke Orhogoal Array Spriger New York 1999 doi:101007/ [11] S Addela Orhogoal Mai Effec Pla for Ay- Copyrigh 01 SciRe

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