The Number of the Two Dimensional Run Length Constrained Arrays

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1 2009 Iteratoal Coferece o Mache Learg ad Coutg IPCSIT vol.3 (20) (20) IACSIT Press Sgaore The Nuber of the Two Desoal Ru Legth Costraed Arrays Tal Ataa Naohsa Otsua 2 Xuerog Yog 3 School of Scece ad Egeerg Toyo De Uversty Hatoyaa-ach Sataa Jaa 2 Dvso of Scece Toyo De UverstyHatoyaa-ach H-gu Sataa Jaa 3 Deartet of Matheatcal Sceces Uversty of Puerto RcoMayaguez PR 0068 USA Abstract. Frst a ew fraewor descrbg the trasfer atrces for the two desoal ru legth costraed arrays (codes) s troduced ad soe ortat roertes of the trasfer atrces T ( ) are derved ths fraewor. The usg these roertes t s show that the ubers N ( )( ) of the two desoal bary arrays satsfyg the - ru legth costrat s exressble by a lear recurrece euato of a fxed order aroxately eual to d( T ) 2. Fally to deostrate the effectveess of the result obtaed soe uercal exales are reseted. Keywords: Ru legth costraed array trasfer atrx forato caacty; data storage. Itroducto Ru legth costraed arrays (codes) are wdely used dgtal data recordg ad trassso. Its geeralzato to the two-desoal case s of otetal terest the age-oreted forato storage techologes such as holograhc storage. A oe-desoal bary seuece s sad to satsfy a ( d )-ru legth costrat f every ru of zeros the seuece has legth at least d ad at ost. Slarly a 2 - desoal bary array s sad to satsfy a ( d )-ru legth costrat f t satsfes the oe-desoal - ru legth costrat both horzotally ad vertcally. Let the uber of 2 -desoal bary arrays of sze satsfyg the ( d )-ru legth costrat be deoted by Nd ( ). The the forato caacty s defed as Cd l log 2 Nd ( ) () whch ay be terreted as the axu uber of bts of forato that ca be stored asytotcally er ut volue. Recetly a great deal of effort has bee ade for evaluatg the ubers Nd ( )( ) ad the lt C d artcular for vestgatg the exstece of the lt ad lower ad uer bouds o Cd for varous values of ( d ). See e.g. []-[6] ad the refereces here. Ths aer focuses o a secal case of the 2-desoal costraed arrays wth d ad. For ths case the uber d : N s well ow as the Fboacc uber. Further t s easly see by exchagg the roles of 0 ad that C C0. For otatoal slcty hereafter N ( ) N0 ( ) wll be deoted sly by N. ( ) Ths secal case was extesvely studed []-[4] ad also [5] ad ay ortat results have bee obtaed. Oe of ortat ad terestg results aog the s the wor by Cal ad Wlf []. I ths wor they used the fact that the uber Nca ( ) be exressed the for t N ( ) T (2) t where dcates the colu vector wth all etres eual to ad a coatble deso deotes the trasose T s the trasfer atrx of the two-desoal - ru legth costrat roble troduced 45

2 []. The trasfer atrx T s a syetrc d d couted the followg recursve aer: d d0 d dd2. atrx wth all etres eual to 0 or ad the order d Therefore all the roertes of Nca ( ) be obtaed through the corresodg trasfer atrxt. Neve ( ) for relatvely large values of ad. Fally to deostrate effectveess of our results the recursve euatos of N ( ) for 234 wll be couted as uercal exales. 2. Descrto of Trasfer Matrces I ths secto we frst troduce a ew fraewor for reresetg the trasfer atrxt defed Cal ad Wlf [] ad vestgate ther ortat roertes. Frst let {0} deote the set of all - desoal atrces wth all etres {0}. The we have the followg results. d LEMMA. Let us troduce the two seueces of atrces V W {0} by the followg recursve forula: () 0 V 0 V 2 V [ d 2 ] ( V ) () 0 0 W W 2 W [ d 2 ] ( W ) where the otato X [ ] dcates the sub-atrx coosed of the frst rows of a atrx X ad 0 s a zero colu vector wth a coatble deso. Further let us rereset the atrces V W as the row vectors: V v w d d {0} W {0} vd w d The the followg stateets hold. () The row vectors v v d are all dstct ad so are ( ) ( w ) w d. () For each d v ad w cocde the reversed order that s f 2 v [ v v v ] the w v v v ( ) w [ ]. () For each d both v ad have o two cosecutve s ther cooets resectvely. (v) Set { v v d } cocdes wth the set of all ossble row vectors v {0} havg o two cosecutve s. Next let be arbtrary tegers ad for ay two atrces defe the two oeratos ad by V W : v w {0} V W : v w {0} v w V {0} W {0} v w f v w 0 f v w where v w : v w : 0 otherwse. 0 otherwse. The the followg theore ca be roved. The stateet () has bee show a dfferet fraewor [6] but t would be worthwhle to rerove t because our fraewor gves a sler ad ore straghtforward roof. s (3) (4) THEOREM. The followg facts hold. () The trasfer atrx T s exressed as. T V V W W 46

3 () Further T s couted by the followg recursve forula: T [ d 2] t T T T [ d 2] 0 T 0. (5) Proof. The stateet () ca be verfed sly by observg that Lea esures that the vectors ( ) ( v ) v d ad ( ) ( w ) w d satsfy the codtos for the trasfer atrx T defed []. To rove () lettg 2 ad assug that () s true for t wll be roved that () s also true for. Ths s doe by drectly evaluatg V ( ) V ( ). LEMMA 2. Cosder two atrces V W ad a erutato atrx P rereseted as v w r V {0} W {0} v w P[ ] {0} where v w are row vectors ad each s a colu ut vector. The the followg eualty holds true: r ( PV ) ( PW ) P( V W ) P. (7) Proof: Frst ote that P s rereseted as a erutato of the detty atrx I [ e e ] where e s the colu ut vector wth -th etry eual to. Therefore t suffces to verfy (6) for a secal case such that P s a erutato atrx of the for P[ e e ] c: where [ ] c:: dcates a terchage of the -th ad -th colu vectors. For ths erutato atrx P we have v w v w ( PV ) ( PW ) Vr: Wr: P( V W ) P v w v w r: r: c: r: where slarly [ ] r: dcates the terchage of the -th ad -th colu vectors. THEOREM 2. Let V W be gve by (4) ad (5) ad defe dd. (8) P : V W {0} The for ay the followg stateets hold. P s a d d syetrc erutato atrx ad satsfes TP PT Recursve Forulas V P W (6). Further I ths secto t s show that the ubers N ( )( ) of the two-desoal bary arrays satsfyg the - ru legth costrat ca be rereseted by a lear recurrece euato of a fxed order. I artcular t s show that the order of the lear recurrece euato s aroxately eual to d( T ) 2 for large values of. LEMMA 3. Let P be gve by (4). The ts trace s gve as Proof. It follows fro (4) that d tr( P ) d ( 2)/2 2 d tr( P ) ( v w ) f s eve f s odd. (9) 47

4 Frst ote that by vrtue of the defto of oerato v w f ad oly f v w. I addto to ths sce by () of Lea v ad w cocde the reversed order therefore f v w the v ust be of the for whe s eve ad whe s odd. v v v 2 v 2 v 2 v 2 v (0) v v v2 v2 v2 v () Furtherore sce v has o two cosecutve s by Lea t s easly see fro (20) ad (2) that a ecessary ad suffcet codto for v w s that v s of the for v v v ( 2) ( 2) 2 v s eve (2) v v v 2 v 2 v 2 v s odd (3) Now (9) s evaluated. Frst for the case of beg eve the suato s eual to the uber of all ossble vectors that has the for of () whch tur the for of v v ( 2) 2 ad hece the suato s eual to d ( 2) 2. Slarly for the case of beg odd the suato (9) s eual to the uber of all ossble vectors that has the for of (2) whch tur the for of v v ad 2 hece the suato s eual to d ( ) 2. Ths coletes the roof. Next let be arbtrarly fxed ad defe a seuece of d -desoal colu vectors by ad a seuece of atrces 2 : : T d () : [ 2 ] [ T T ]. (4) ( d) LEMMA 4. The followg stateets hold: () The vectors defed (3) satsfy P () The atrces defed (4) satsfy ra ra ra ra 2. Proof. The stateet () ca be easly roved usg Theore 2 (). I fact for ay P P T T P T. Next to rove () assue ra ra. The frst ote that ra ra les that ca be exressed as a lear cobato of the vectors.e. for soe vector R. Ths fact leads to T T T [ ] [ ] [ e e ]. 0 2 Therefore s also a lear cobato of 0 ad hece ra rara 2. Reeatg ths rocess oe ca verfy ra ra for all 2. LEMMA 5. Let be the atrces defed by (4). The ax ra { d tr( P)} : r 0 2 Proof. Frst recall that P s a syetrc erutato atrx ad tr( P ) s gve by (8). It s ot dffcult to see fro LEMMA 4 () that for every tr( P ) etres the vector are fxed by ths erutato P ad the other etres are ar-wsely eruted but the two etres each ar are eual. That s to say there exst two subsets { } ad { } of { d } wth : tr( P ) ad : ( d )/2 deedet of such that the etres of are ot eruted by P ad 48

5 the other etres are ar-wsely eual.e.. Therefore the row vectors of gve (4) satsfy ( ) ( ) ( ) ( ) 0. Therefore there are at least learly deedet vectors { () ( d )} ad hece ax ra d {d tr( P)} {d tr( P)} whch roves the desred result Now t s ready to state ad rove our a theore as follows. THEOREM 3. Let be arbtrarly fxed. Cosder the ubers Ngve ( ) by (2) ad the vectors gve by (4) that s The the followg stateets hold: t N ( ) T : T. () The seuece { } satsfes a lear recurrece euato of order r that s r where ( r ) are soe costats deedet of. 2 2 r r t N T () The ubers ( ) ( ) also satsfy the sae lear recurrece euato as (28) that s N ( ) N ( ) N ( 2) N ( r) r (6) 2 r Proof. Ths theore s easly roved. I fact by vrtue of LEMMA 4 wth r s exressed as a lear recursve euato of the for (28) whch ca be also wrtte as ad sce 2 3 r T Td 2 T r T r t the lear recurrece euato (6) drectly follows for (30). N ( ) T ( ) 4. Refereces (5) (7) [] N. Cal ad H. Wlf The uber of deedet sets a grd grah SIAM Joural o Dscrete Matheatcs vol Feb [2] W. Wees ad R. E. Blahut The caacty ad codg ga of certa checerboard codes IEEE Tras. Ifor. Theory vol May 998. [3] S. Forchhaer ad J. Justese Etroy bouds o the caacty of 2-desoal rado felds IEEE Tras. Ifor. Theory vol Ja [4] Kato ad K. Zeger O the caacty of two-desoal ru legth costraed chaels IEEE Tras. Ifor. Theory vol July 999. [5] Z. Nagy ad K. Zeger Caacty bouds for the 3-desoal (0) ru legth lted chael IEEE Tras. Ifor. Theory vol May [6] X. R. Yog The chael caacty of oe ad two-desoal costraed codes Ph. D Thess Hog Kog Uversty of Scece ad Techology

for each of its columns. A quick calculation will verify that: thus m < dim(v). Then a basis of V with respect to which T has the form: A

for each of its columns. A quick calculation will verify that: thus m < dim(v). Then a basis of V with respect to which T has the form: A Desty of dagoalzable square atrces Studet: Dael Cervoe; Metor: Saravaa Thyagaraa Uversty of Chcago VIGRE REU, Suer 7. For ths etre aer, we wll refer to V as a vector sace over ad L(V) as the set of lear