Frequency hopping sequences with optimal partial Hamming correlation

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1 1 Frqucy hoppg squcs wth optmal partal Hammg corrlato Jgju Bao ad ju J arxv: v2 [cs.it] 11 Nov 2015 Abstract Frqucy hoppg squcs (FHSs) wth favorabl partal Hammg corrlato proprts hav mportat applcatos may sychrozato ad multpl-accss systms. I ths papr, w vstgat costructos of FHSs ad FHS sts wth optmal partal Hammg corrlato. W frst stablsh a corrspodc btw FHS sts wth optmal partal Hammg corrlato ad multpl partto-typ balacd std cyclc dffrc packgs wth a spcal proprty. By vrtu of ths corrspodc, som FHSs ad FHS sts wth optmal partal Hammg corrlato ar costructd from varous combatoral structurs such as cyclc dffrc packgs, ad cyclc rlatv dffrc famls. W also dscrb a drct costructo ad two rcursv costructos for FHS sts wth optmal partal Hammg corrlato. As a cosquc, our costructos yld w FHSs ad FHS sts wth optmal partal Hammg corrlato. Idx Trms Frqucy hoppg squcs (FHSs), partal Hammg corrlato, partto-typ cyclc dffrc packgs, cyclc rlatv dffrc famls, cyclotomy. I. INTRODUCTION FREQUENCY hoppg (FH) multpl-accss s wdly usd th modr commucato systms such as ultrawdbad (UWB), mltary commucatos, Blutooth ad so o, for xampl, [5], [17], [29]. I FH multplaccss commucato systms, frqucy hoppg squcs ar mployd to spcfy th frqucy o whch ach sdr trasmts a mssag at ay gv tm. A mportat compot of FH sprad-spctrum systms s a famly of squcs havg good corrlato proprts for squc lgth ovr sutabl umbr of avalabl frqucs. Th optmalty of corrlato proprty s usually masurd accordg to th wll-kow mpl-grbrgr boud ad Pg-Fa bouds. Durg ths dcads, may algbrac or combatoral costructos for FHSs or FHS sts mtg ths bouds hav b proposd, s [7]-[11], [14]-[15], [18], [20]-[21], [0]- [], ad th rfrcs thr. Compard wth th tradtoal prodc Hammg corrlato, th partal Hammg corrlato of FHSs s much lss wll studd. Nvrthlss, FHSs wth good partal Hammg corrlato proprts ar mportat for crta applcato scaros whr a approprat wdow lgth shortr tha th total prod of th squcs s chos to mmz th sychrozato tm or to rduc th hardwar complxty of Ths work was supportd by th NSFC udr Grats , , ad a projct fudd by th prorty acadmc program dvlopmt of Jagsu hghr ducato sttutos. J. Bao ad. J ar wth th Dpartmt of Mathmatcs, Soochow Uvrsty, Suzhou , P. R. Cha. E-mal: baojgju@hotmal.com; jlju@suda.du.c. th FH-CDMA rcvr [16]. Thrfor, for ths stuatos, t s cssary to cosdr th partal Hammg corrlato rathr tha th full-prod Hammg corrlato. I 2004, Eu t al. [16] gralzd th mpl-grbrgr boud o th prodc Hammg autocorrlato to th cas of partal Hammg autocorrlato, ad obtad a class of FHSs wth optmal partal autocorrlato [28]. I 2012, Zhou t al. [4] xtdd th Pg-Fa bouds o th prodc Hammg corrlato of FHS sts to th cas of partal Hammg corrlato. Basd o m-squcs, Zhou t al. [4] costructd both dvdual FHSs ad FHS sts wth optmal partal Hammg corrlato. Vry rctly, Ca t al. [6] mprovd lowr bouds o partal Hammg corrlato of FHSs ad FHS sts, ad basd o gralzd cyclotomy, thy costructd FHS sts wth optmal partal Hammg corrlato. I ths papr, w prst som costructos for FHSs ad FHS sts wth optmal partal Hammg corrlato. Frst of all, w gv combatoral charactrzatos of FHSs ad FHS sts wth optmal partal Hammg corrlato. Scodly, by mployg partto-typ balacd std cyclc dffrc packgs, cyclc rlatv dffrc famls, ad cyclc rlatv dffrc packgs, w obta som FHSs ad FHS sts wth optmal partal Hammg corrlato. Fally, w prst two rcursv costructos for FHS sts, whch cras thr lgths ad alphabt szs, ad prsrv thr optmal partal Hammg corrlatos. Our costructos yld optmal FHSs ad FHS sts wth w ad flxbl paramtrs ot covrd th ltratur. Th paramtrs of FHSs ad FHS sts wth optmal partal Hammg corrlato from th kow rsults ad th w os ar lstd th tabl. Th rmadr of ths papr s orgazd as follows. Scto II troducs th kow bouds o th partal Hammg corrlato of FHSs ad FHS sts. Scto III prsts combatoral charactrzatos of FHSs ad FHS sts wth optmal partal Hammg corrlato. Scto IV gvs som combatoral costructos of FHSs ad FHS sts wth optmal partal Hammg corrlato by usg partto-typ balacd std cyclc dffrc packgs, cyclc rlatv dffrc famls ad cyclotomc classs. Scto V prsts a drct costructo of FHS sts wth optmal partal Hammg corrlato. Scto VI prsts two rcursv costructos of FHS sts wth optmal partal Hammg corrlato. Scto VII cocluds ths papr wth som rmarks. II. OWER BOUNDS ON THE PARTIA HAMMING CORREATION OF FHSS AND FHS SETS I ths scto, w troduc som kow lowr bouds o th partal Hammg corrlato of FHSs ad FHS sts.

2 2 KNOWN AND NEW FHSs WITH OPTIMA PARTIA HAMMING CORREATION gth Alphabt H max ovr corrlato Numbr of sz wdows of lgth squcs Costrats Sourc q 2 1 q+1 q 1 [16] q m 1 q m 1 (q 1) q m 1 1 [4] q m 1 q m 1 (q 1) d (q 1), d q m 1 d gcd(d,m) 1 [4] v v v f [6] 2v 2v +1 v v 2v+1 2v 8v+1 8v 2v+1 2v v+1 v 4 4v+2 4v 6v 2v +1 p(p m 1) p m v v v 4v 16v v 2v v p m 1 1 Thorm Thorm p 1 (mod 12) s a prm Thorm p 1 (mod 6) s a prm Thorm p 1 (mod 12) s a prm Thorm p 1 (mod 4) s a prm Thorm p 7 (mod 12) s a prm Thorm p 5 (mod 8) s a prm Thorm 6.5 p m 1 m 2 Thorm 5.1 vw (v 1)w+ w r vw f q 1 p 1 > 2, v > ( 2) ad gcd(w,) 1 v qm 1 vq m 1 (q 1) m > 1, ad q m p 1 d v(q m 1) d q m 1 d p 1 for 1 s q s a prm powr; v s a tgr wth prm factor dcomposto v p m1 1 pm2 2 p ms s wth p 1 < p 2 <... < p s ;,f ar tgrs such that > 1 ad gcd(p 1 1,p 2 1,...,p s 1), ad f p1 1 ; w s a tgr wth prm factor dcomposto w q 1 1 q2 2 qt t wth q 1 < q 2 <... < q t ; r s a tgr such that r > 1 ad r gcd(,q 1 1,q 2 1,...,q t 1); p s a prm; d, m ar postv tgrs. Corollary 6.6 Corollary 6.11 For ay postv tgr l 2, lt F {f 0,f 1,...,f l 1 } b a st of l avalabl frqucs, also calld a alphabt. A squc X {x(t)} t0 1 s calld a frqucy hoppg squc (FHS) of lgthovrf f x(t) F for all 0 t 1. For ay two FHSs X {x(t)} t0 1 ad Y {y(t)} 1 t0 of lgth ovr F, th partal Hammg corrlato fucto of X ad Y for a corrlato wdow lgth startg at j s dfd by H X,Y (τ;j ) j+ 1 tj h[x(t),y(t+τ)],0 τ <, (1) whr,j ar tgrs wth 1, 0 j <, h[a,b] 1 f a b ad 0 othrws, ad th addto s prformd modulo. I partcular, f, th partal Hammg corrlato fucto dfd (1) bcoms th covtoal prodc Hammg corrlato [24]. If x(t) y(t) for all 0 t 1,.., X Y, w call H X,X (τ;j ) th partal Hammg autocorrlato of X; othrws, w say H X,Y (τ;j ) th partal Hammg cross-corrlato ofx ad Y. For ay two dstct squcs X,Y ovr F ad gv tgr 1, w df ad H(X;) max 0 j< max 1 τ< {H X,X(τ;j )} H(X,Y;) max 0 j< max 0 τ< {H X,Y(τ;j )}. That s, H(X; ) dots th maxmum partal Hammg autocorrlato of X alog wth a arbtrary corrlato wdow of lgth, ad H(X,Y;) dots th maxmum partal Hammg cross-corrlato of X ad Y alog wth a arbtrary corrlato wdow of lgth. For ay FHS of lgth ovr a alphabt of sz l ad ach wdow lgth wth 1, Eu tal. [16] ( ǫ)(+ǫ l) l( 1) drvd a lowr boud: H(X; ), whr ǫ s th last ogatv rsdu of modulo l, whch s a gralzato of th mpl-grbrgr boud [24]. Rctly, such a lowr boud was mprovd by Ca t al. [6] as follows.

3 mma 2.1: ([6]) t X b a FHS of lgth ovr a alphabt of sz l. Th, for ach wdow lgth wth 1, H(X;) ( ǫ)(+ǫ l) l( 1), (2) whr ǫ s th last ogatv rsdu of modulo l. Rcall that th corrlato wdow lgth may chag from cas to cas accordg to th chal codtos practcal systms. Hc, t s vry dsrabl that th volvd FHSs hav optmal partal Hammg corrlato for ay wdow lgth. Th followg dfto s orgatd from th trmology of strctly optmal FHSs [16]. Dfto 2.2: t X b a FHS of lgth ovr a alphabt F. It s sad to b strctly optmal or a FHS wth optmal partal Hammg corrlato f th boud mma 2.1 s mt for a arbtrary corrlato wdow lgth wth 1. Wh, th boud mma 2.1 s xactly th mpl-grbrgr boud [24]. It s clar that ach strctly optmal FHS s also optmal wth rspct to th mpl- Grbrgr boud, but ot vc vrsa [16]. t S b a st of M FHSs of lgth ovr a alphabt F of sz l. For ay gv corrlato wdow lgth, th maxmum otrval partal Hammg corrlato H(S; ) of th squc st S s dfd by H(S;) max{max H(X;), max H(X,Y;)}. X S X,Y S, X Y Throughout ths papr, w us (,M,;l) to dot a st S of M FHSs of lgth ovr a alphabt F of sz l, whr H(S;), ad w us (,;l) to dot a FHS X of lgth ovr a alphabt F of sz l, whr H(X;). Wh, Pg ad Fa [26] dscrbd th followg bouds o H(S; ), whch tak to cosdrato th umbr of squcs th st S. mma 2.: ([26]) t S b a st of M squcs of lgth ovr a alphabt F of sz l. Df I M l. Th (M l) H(S;) (M 1)l ad 2IM (I +1)Il H(S;). (M 1)M I 2012, Zhou t al. [4] xtdd th Pg-Fa bouds to th cas of th partal Hammg corrlato. Thy obtad H(S; ) (M l) (M 1)l ad H(S; ) 2IM (I+l)Il (M 1)M. Rctly, such lowr bouds wr mprovd by Ca t al. [6]. mma 2.4: ([6]) t S b a st of M FHSs of lgth ovr F of sz l. Df I M l. Th, for a arbtrary wdow lgth wth 1, (M l) H(S;) () (M 1)l ad H(S;) 2IM (I +1)Il (M 1)M. (4) Paralll to th oto of strctly optmal dvdual FHSs, Ca t al. gav th followg dfto of strctly optmal FHS sts [6]. Dfto 2.5: A FHS st S s sad to b strctly optmal or a FHS st wth optmal partal Hammg corrlato f o of th bouds mma 2.4 s mt for a arbtrary corrlato wdow lgth wth 1. Not that, wh, th bouds mma 2.4 ar xactly th Pg-Fa bouds. It turs out that ach strctly optmal FHS st s also optmal wth rspct to th Pg-Fa bouds, but ot vc vrsa. III. COMBINATORIA CHARACTERIZATIONS A. A combatoral charactrzato of Strctly optmal FHSs A covt way of vwg FHSs s from a st-thortc prspctv. I 2004, Fuj-Hara t al. [18] rvald a cocto btw FHSs ad partto-typ cyclc dffrc packgs. I ths subscto, w gv a combatoral charactrzato of strctly optmal FHSs. Throughout ths papr w always assum that I l {0,1,2,...,l 1} ad Z s th rsdual-class rg of tgrs modulo. For tgrs s,, th otato s mas that s dvsbl by s. For A Z, τ Z ad a postv tgr, A+τ s dfd to b {a+τ : a A} ad [A] 1 A, whr s th multst uo. A (,;l)-fhs, X (x(0),x(1),...,x( 1)), ovr a frqucy lbrary F ca b trprtd as a st B of l sts B 0,B 1,...,B l 1 such that ach st B corrspods to frqucy F ad th lmts ach st B spcfy th posto dcs th FHS X at whch frqucy appars. Th corrlato proprty ca b rphrasd ths st-thortc framwork. As statd [18, mma 2.1], th st B s a partto of Z ad H X,X (τ;0 ) l 1 B (B +τ) 0 0 for 1 τ <,.., B satsfs th followg proprty trms of dffrcs: ach τ Z \ {0} ca b rprstd as a dffrc b b, b,b B, 0 l 1, at most ways. Ths obsrvato rvals a cocto btw FHSs ad combatoral structurs calld partto-typ cyclc dffrc packgs. Assocatd wth a o-mpty subst B Z, th dffrc lst of B from combatoral dsg thory s dfd to b th multst (B) {a b : a,b B ad a b}. For ay famly B {B 0,B 1,...,B l 1 } of l o-mpty substs (calld bas blocks) of Z, df th dffrc lst of B to b th uo of multsts (B) I l (B ). If th dffrc lst (B) cotas ach o-zro rsdu of Z at most tms, th B s sad to b a (,K,)-CDP (cyclc dffrc packg), whr K { B : I l }. Wh K {k}, w smply wrt k stad of {k}. Th umbr l of th bas blocks B s rfrd to as th sz of th CDP.

4 4 If th dffrc lst (B) cotas ach o-zro rsdu of Z xactly tms, th B s a cyclc dffrc famly ad dotd by (, K, )-CDF. A (,K,)-CDP B {B 0,B 1,...,B l 1 } s calld a partto-typ cyclc dffrc packg f vry lmt of Z s cotad xactly o block B. I 2004, Fuj-Hara t al. [18] rvald a cocto btw FHSs ad partto-typ cyclc dffrc packgs as follows. Thorm.1: ([18]) Thr xsts a (,;l)-fhs ovr a frqucy lbrary F f ad oly f thr xsts a partto-typ (,K,)-CDP of sz l, B {B 0,B 1,...,B l 1 } ovr Z, whr K { B : 0 l 1}. Fuj-Hara t al. [18] also gav a smplfd vrso of th mpl-grbrgr boud as follows. mma.2: ([18]) For a arbtrary (,;l)-fhs, t holds that { k f l, ad 0 f l, whr kl + ǫ, 0 ǫ l 1. Ths mpls that wh > l, th squc s optmal f k. mma.: ([18]) t kl + l 1 wth k 1. Th thr xsts a optmal (,k;l)-fhs f ad oly f thr xsts a partto-typ (,{k,k+1},k)-cdf whch l 1 blocks ar of sz k +1 ad th rmag o s of sz k. For a u-tupl T (a 0,a 1,...,a u 1 ) ovr Z, th multst (T) {a j+ a j : 0 j u 1} s calld -apart dffrc lst of th tupl, whr 1 u, j + s rducd modulo u, ad a j+ a j s tak as th last postv rsdu modulo. For a partto-typ CDP of sz l ovr Z, B {B 0,B 1,...,B l 1 }, for 1 τ 1, lt D(τ) {a : 0 a <,{a, a+τ} B,B B}, D(τ) (a0,a 1,...,a u 1 ),whr 0 a 0 < < a u 1 < ad {a 0,a 1,...,a u 1 } D(τ),ad d B m 1 τ< m{g : g {} ( D(τ))}, for 1 max{ D(τ) : 1 τ < }, whr ( D(τ)) f D(τ) or > D(τ). W call D(τ) th orbt cycl of τ B ad d B th mmal -apart dstac of all orbts. Not that D( τ) D(τ)+τ (mod ) ad m{g : g {} ( D(τ))} m{g : g {} ( D( τ))}. W llustrat th dfto of d B th followg xampl. Exampl.4: I ths xampl, w costruct a partto-typ (0,11,2)-CDP, B, wth d B 15 for 0 < 2. St B {{1,2,14},{,7,9},{4,2,26},{6,1,27}, {16,17,29},{18,22,24},{8,11,19},{12,21,28}, {0,5},{15,20},{10,25}}. It s radly chckd that B s a (0,11,2)-CDP. By th fact m{g : g {} ( D(τ))} m{g : g {} ( D( τ))}, w oly d to comput D(τ) for1 τ 15, so as to obta d B. Smpl computato shows that D(1) (1,16), D(2) (7,22), D() (8,2), D(4) (,18), D(5) (0,15), D(6) (,18), D(7) (6,21), D(8) (11,26),D(9), D(10), D(11) (8,2), D(12) (2,17), D(1) (1,16), D(14) (1,28), D(15) (10,25). Th, d B 15 for 0 < 2. By drct chck or by Thorm.5, th corrspodg FHS s a strctly optmal (0, 2; 11)- FHS. W ar a posto to gv a combatoral charactrzato of strctly optmal FHSs. Thorm.5: Thr s a strctly optmal (, l ;l)-fhs wth rspct to th boud (2) f ad oly f thr xsts a partto-typ (,K, l )-CDP of sz l, B {B 0,B 1,...,B l 1 } ovr Z, such that d B for l 1 l, whr K { Br : 0 r l 1}. Proof: t X b a FHS of lgth ovr F I l. t B r {t : x(t) r,0 t 1} for ach r F. Th B r s th st of posto dcs th squc X (x(0),...,x( 1)) at whch th frqucy r appars. By Thorm.1, th squc X s a (, l ;l)-fhs ovr F f ad oly f B {B 0,B 1,...,B l 1 } s a partto-typ (,K, l )-CDP of sz l, whr K { Br : 0 r l 1}. It s lft to show that X s strctly optmal wth rspct to th boud (2) f ad oly f d B for 1 l. l Dot l. By mma 2.1, X s strctly optmal wth rspct to boud (2) f ad oly f H(X;) for 1,.., for 1, th qualty H(X;) holds for ( 1) <. W show that th qualty H(X;) holds for ( 1) < f ad oly f d B. By th dfto of H X,X (τ;j ) ad D(τ), w hav H X,X (τ;j ) D(τ;j ), whr D(τ;j ) D(τ) {j,j + 1,j + 2,...,j + 1} ad th arthmtc j + t (1 t 1) s rducd modulo. Dot d (τ) m{g : g {} ( D(τ))}. Clarly, max { D(τ;j ) } f ad oly f d (τ). Hc, 0 j< H(X;) max max 1 τ<0 j< { D(τ;j ) } f ad oly f d (τ) for 1 τ <,.., d B m{d (τ) : 1 τ < }. It follows that H(X;) for ( 1) < f ad oly f d B. Ths complts th proof. A (mg, g, K, )-cyclc rlatv dffrc packg (brfly CRDP) s a (mg,k,)-cdp ovr Z mg, B {B : I u }, such that o lmt of mz mg occurs (B), whr mz mg {0,m,...,(g 1)m}. Th umbr u of th bas blocks B s rfrd to as th sz of th CRDP. If (B) cotas ach lmt of Z mg \mz mg xactly tms ad o lmt of mz mg occurs, th B s calld a (mg,g,k,)- cyclc rlatv dffrc famly (brfly CRDF).

5 5 O mportac of a CRDP s that w ca put a approprat CDP o ts subgroup to drv a w CDP. W stat a smpl but usful fact th followg lmma. mma.6: t B b a (mg, g, K, 1)-CRDP of sz u ovr Z mg rlatv to mz mg, whos lmts of bas blocks, togthr wth 0,m,...,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg, whr s g. t A b a partto-typ (g,k, g s )-CDP of sz r ovr Z g wth d A s for 1 g s. Th thr xsts a partto-typ (mg,k K, g s )-CDP of sz gu s +r, D such that dd sm for 1 g s. Proof: t B {B + jsm : B B,0 j < g s }, A {ma : A A} ad D B A. Sc B s a (mg,g,k,1)-crdp of sz u ovr Z mg rlatv to mz mg ad all lmts of bas blocks, togthr wth 0,m,2m,...,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg, w hav that (B) Z mg \mz mg ad B s a partto of Z mg \mz mg. Clarly, (B ) cotas ach τ (B) xactly g s tms ad o lmt of Z mg \ (B). Sc A s a partto-typ (g,k, g s )-CDP of sz r ovr Z g, w gt (ma) m (A) ada s a partto ofmz mg. A A Hc, D s a partto-typ (mg,k K, g s )-CDP of sz gu s +r. Clarly, f τ (B), th th orbt cycl D(τ) of τ D s of th form (a 0,a 0 +sm,a 0 +2sm,...,a 0 +mg sm), ad f τ (A ), th th orbt cycl D(τ) of τ D s of th form (mb 0,mb 1,...,mb s 1), whr (b 0,b 1,...,b s 1) s th orbt cycl of τ m A, othrws D(τ). Hc, dd (τ) sm for τ (B), d D (τ) sm for τ (A ) bcaus d A s for 1 g s, ad dd (τ) mg for τ / (D), whr d D (τ) m{g : g {mg} ( D(τ))}. It follows that d D m{d D (τ) : 0 < τ < mg} sm for 1 g s. Thrfor, D s th rqurd partto-typ (mg,k K, g s )- CDP. Ths complts th proof. t B b a (mg,g,k,1)-crdf wth k 1 g, f all lmts of bas blocks, togthr wth 0,m,...,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg, whr s g k 1, th ths (mg,g,k,1)-crdf s sad to b rsolvabl. Thr ar svral costructos for rsolvabl (mg, g, k, 1)-CRDFs []-[4]. For xampl, th st B {{1,2,14},{,7,9},{4,2,26}, {6,1,27}} s a rsolvabl (0,6,,1)-CRDF ovr Z 0. It s asy to s that A {{0,1},{,4},{2,5}} s a (6,2,2)- CDP wth d A for 1 2. By mma.6, w also obta a (0,{2,},2)-CDP of sz 11 such that d 15 for 1 2, whch s th sam as that Exampl.4. By vrtu of th combatoral charactrzato of strctly optmal FHSs, th xstc of a strctly optmal(kv,k; kv+1 k+1 )- FHS s quvalt to th xstc of a rsolvabl (kv,k,k+ 1,1)-CRDF. Thorm.7: t k ad v b postv tgrs wth k+1 v 1. Th thr xsts a strctly optmal(kv,k; kv+1 k+1 )-FHS f ad oly f thr xsts a rsolvabl (kv,k,k +1,1)-CRDF. Proof: : By mma. ad Thorm.5, thr s a strctly optmal (kv,k; kv+1 k+1 )-FHS f ad oly f thr s a partto-typ (kv,{k,k +1},k)-CDF ovr Z kv wth d v for 1 k, whch kv k k+1 blocks ar of sz k + 1 ad th rmag o s of sz k. W shall prov that such a partto-typ (kv,{k, k + 1}, k)-cdf xsts f ad oly f thr s a rsolvabl (kv,k,k + 1,1)-CRDF rlatv to vz kv {0,v,...,(k 1)v}. tb b such a partto-typ(kv,{k,k+1},k)-cdf whr B {B 0,B 1,...,B (kv k)/(k+1) }. By th dfto of CDF, for 1 τ kv 1, ach dffrc τ occurs xactly k tms, thus D(τ) (a 0,a 1,...,a k 1 ) s a k-tupl vctor of dstct lmts. Clarly, d 1 (τ) m{g : g {kv} 1 ( D(τ))} kv k v. Sc d v, w hav d 1 (τ) v. Ths lads to a j+1 a j +v for 0 j k 1. Hc, for ay par {a,b} cotad som bas block of B, th par {a+v,b+v} s also cotad a bas block of B. Sc B s a partto-typ CDF, th st B j +v s also a bas block of B ad o bas block of B s of th form vz kv +t. Wthout loss of gralty, lt B (kv k)/(k+1) vz kv +t ad B j +v B j+ v 1 for 0 k+1 j < (kv k)/(k+1), whr j+ v 1 kv k k+1 s rducd modulo k+1. It s asy to s that {B 0 t,b 1 t,...,b (v k 2)/(k+1) t} s th st of bas blocks of a rsolvabl (kv,k,k+1,1)-crdf rlatv to vz kv. Covrsly, suppos that thr s a rsolvabl (kv,k,k + 1,1)-CRDF ovr Z kv rlatv to vz kv. Th s k k 1. Sc A {Z k } s th trval partto-typ (k,k,k)-cdf wth d A for 1 k, applyg mma.6 ylds a partto-typ (kv,{k,k +1},k)-CDF of sz kv+1 k+1 that d D v for 1 k. Ths complts th proof., D, such B. A combatoral charactrzato of strctly optmal FHS sts I 2009, G t al. [21] rvald a cocto btw FHS sts ad famls of partto-typ balacd std cyclc dffrc packgs. I ths subscto, w xtd th combatoral approach usd Scto III-A for dsgg strctly optmal FHSs to gv a combatoral charactrzato of strctly optmal FHS sts. t S b a (,M,;l)-FHS st ovr F I l. For X S, lt Br X {t : x(t) r,0 j 1} for ach r F. Th Br X s th st of posto dcs th squc X (x(0),...,x( 1)) at whch th frqucy r appars. For two dstct FHSs X,Y S, t s asy to s that th Hammg cross-corrlato H X,Y (τ;0 ) l 1 (B X +τ) B Y for 0 τ <,.., {BX : I l } ad {B Y : I l } satsfy th followg proprty trms of xtral dffrcs: ach τ Z ca b rprstd as a dffrc b b, (b,b ) B X B Y, 0 l 1, at most ways. Ths obsrvato rvals a cocto btw FHS sts ad combatoral structurs calld famls of partto-typ balacd std cyclc dffrc packgs. ta,b b two substs ofz v. Th lst of xtral dffrc of ordrd par (A,B) s th multst 0 E (A,B) {y x : (x,y) A B}. Not that th lst of xtral dffrc E (A,B) may cota zro. For ay rsdu τ Z v, th umbr of occurrcs of τ E (A,B) s clarly qual to (A+τ) B.

6 6 t B j,0 j M 1, b a collcto of l substs B j 0,...,Bj l 1 of Z, rspctvly. Th lst of xtral dffrc of ordrd par (B j,b j ), 0 j j < M, s th uo of multsts E (B j,b j ) E (B j,bj ). I l If ach B j s a (,K j,)-cdp of sz l, ad E (B j,b j ) cotas ach rsdu of Z at most tms for 0 j j < M, th th st {B 0,...,B M 1 } of CDPs s sad to b balacd std wth dx ad dotd by (,{K 0,...,K M 1 },)-BNCDP. If ach B j s a parttotyp CDP for 0 j < M, th th (,{K 0,...,K M 1 },)- BNCDP s calld partto-typ. For covt, th umbr l of th bas blocks B j s also sad to b th sz of th BNCDP. I 2009, G t al. [21] rvald a cocto btw FHS sts ad partto-typ BNCDPs as follows. Thorm.8: ([21]) Thr xsts a (,M,;l)-FHS st ovr a frqucy lbrary F f ad oly f thr xsts a partto-typ (,{K 0,K 1,...,K M 1 },)-BNCDP of sz l. t B {B X : X S} b a famly of M partto-typ CDPs of sz l ovr Z, whr B X {B0 X,BX 1,...,BX l 1 },X S. For two dstct partto-typ CDPs B X, B Y, ad for 0 τ 1, lt D (X,Y) (τ) {a : 0 a <, (a, a+τ) B X B Y, for som I l }, D (X,Y) (τ) (a 0,a 1,...,a u 1 ), d (X,Y) whr 0 a 0 < < a u 1 < ad {a 0,a 1,...,a u 1 } D (X,Y) (τ), m m{g : g {} ( D (X,Y) (τ))}, 0 τ< for 0 max{ D (X,Y) (τ) : 0 τ < },ad d B m{m X Y {d(x,y) },m X S {dx }}. whr ( D (X,Y) (τ)) f D (X,Y) (τ) or > D (X,Y) (τ). If X Y, th D (X,X) (τ), D (X,X) (τ) ad ar th sam as D(τ), D(τ) ad d X, rspctvly. Not that D (Y,X) ( τ) D (X,Y) (τ) + τ (mod ) ad m{g : g {} ( D (X,Y) (τ))} m{g : g d (X,X) {} ( D (Y,X) ( τ))}. W llustrat th dfto of d B of a st of CDPs th followg xampl. Exampl.9: I ths xampl, w costruct a partto-typ (24,{{2,},{2,},{2,}},)-BNCDP, B, wth d B 8 for 1. St B j j {4,8}, Bj 1+j {0,20}, Bj 2+j {12,16}, B j +j {9,22,2},Bj 4+j {1,14,15},Bj 5+j {6,7,17}, B j 6+j {2,19,21},Bj 7+j {11,1,18},Bj 8+j {,5,10}, whr j {0, 1, 2} ad th addto th subscrpt s prformd modulo 9. t B j {B j r : 0 r 8} for j {0,1,2} ad B {B 0,B 1,B 2 }. Clarly, (B j ) [{±1,±2,±4,±5,±10,±11,±17}], E(B 0,B 1 ) E(B 0,B 2 ) E(B 1,B 2 ) [Z 24 \{0,8,16}]. Th B s a partto-typ (24,{{2,},{2,},{2,}},)- BNCDP. For achτ (B j ),0 j <, th orbt cycl D (Bj,B j)(τ) of τ s of th form (a,a + 8,a + 16) for som a, for xampl, D (B j,b j )(4) (4,12,20). For τ E (B k,b j ), 0 k < j 2, th orbt cycl D (B k,b j)(τ ) of τ s also of th form (a,a+8,a+16) for som a, for xampl, D (B 0,B 1 )(22) (4,12,20). Thrfor, ( D (Bj,B j )(τ)) ( D (B k,b j )(τ )) 8 for 1. So, for 0 k,j 2 ad 1, w hav d (Bk,B j ) m m{g : g {24} ( D (B 0 τ<24 k,b j )(τ))} 8, d B m{ m k 0 k<j 2 {d(b,b j ) j 0 j 2 {d(b,b j ) }, m }} 8. By drct chck or by Thorm.10, th corrspodg FHS st s a strctly optmal (24,,;9)-FHS st wth rspct to th boud (4). Th followg thorm clarfs th cocto btw strctly optmal FHS sts ad partto-typ BNCDPs wth a spcal proprty. Thorm.10: Thr s a strctly optmal (,M,;l)-FHS st S wth rspct to th boud (4) f ad oly f thr xsts a partto-typ (,{K X : X S},)-BNCDP of sz l, B {B X {B0 X,BX 1,...,BX l 1 } : X S} ovr Z wth d B l 1}, for 1, whr KX { Br X : 0 r 2IM (I+1)Il (M 1)M ad I M l. Proof: t S b a st of M squcs of lgth ovr F I l. t Br X {t : x(t) r,0 t 1} for ach r F ad X S. Th Br X s th st of posto dcs th squc X (x(0),...,x( 1)) at whch th frqucy r appars. By Thorm.8, th st S of M squcs s a (,M,;l)-FHS st ovr F f ad oly f {B X {B0 X,...,BX l 1 } : X S} s a partto-typ (,{K X : X S},)-BNCDP, whr K X { Br X : 0 r l 1}. It s lft to show that S s strctly optmal wth rspct to th boud (4) f ad oly f d B for 1, whr B {B X : X S}. By mma 2.4, S s strctly optmal wth rspct to th boud (4) f ad oly f H(S;) for 1,.., for 1, th qualty H(S;) holds for ( 1) <. W show that th qualty H(S;) holds for ( 1) < f ad oly f d B. By th dfto of H X,Y (τ;j ) ad D (X,Y) (τ), w hav H X,Y (τ;j ) D (X,Y) (τ;j ), whr 0 < τ < f X Y, 0 τ < f X Y, D (X,Y) (τ;j ) D (X,Y) (τ) {j,j + 1,...,j + 1} ad th arthmtc j + t (1 t 1) s rducd modulo. Dot d (X,Y) (τ) m{g : g {} ( D (X,Y ) (τ))}. Th, max { D (X,Y)(τ;j ) } f ad oly d (X,Y ) (τ). 0 j< Hc, H(X,Y;) max max { D (X,Y)(τ;j ) } f τ 0 j<

7 7 ad oly f d (X,Y) (τ) for 0 τ < f X Y ad for 1 τ < f X Y,.., d (X,Y) m{d (X,Y ) τ (τ)}. It follows that H(X,Y;) for ( 1) < (X,Y) f ad oly f d. Hc, H(S;) max {H X,X (τ;j )}, max {H X,Y (τ;j )}} { max 0 j< X S for 1 τ<, ( 1) 0 τ<, X Y f ad oly f d (X,Y ), < for X,Y S, whch s quvalt to d B m{d (X,Y) X,Y S}. Ths complts th proof. : t B j b a (mg,g,k j,1)-crdp ovr Z mg for 0 j < M, whr B j {B j 0,Bj 1,...,Bj u 1 }. Th st {B 0,...,B M 1 } s rfrrd to as a (mg,g,{k 0,K 1,...,K M 1 },1)-BNCRDP (balacd std cyclc rlatv dffrc packg) ovr Z mg, f (B j,b j ) cotas ach lmt of Z mg \mz mg at most oc ad o lmt of mz g occurs for 0 j j < M. For covt, th umbr u of th bas blocks B j s also sad to b th sz of th BNCDP. O mportac of a BNCRDP s that w ca put a approprat BNCDP o ts subgroup to drv a w BNCDP. mma.11: Suppos that thr xsts a (mg,g,{k 0,...,K M 1 },1)-BNCRDP of sz u, {B 0,...,B M 1 }, such that all lmts of bas blocks of B j, togthr wth 0,m,...,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg for 0 j < M, whr s g. If thr xsts a partto-typ (g,{k 0,...,K M 1 }, g s )-BNCDP of sz l, A such that d A s for 1 g s. Th thr xsts a partto-typ (mg,{k 0 K 0,...,K M 1 K M 1 }, g s )-BNCDP of sz gu s +l, D such that dd sm for 1 g s. Proof: Dot B j {B j k : 0 k < u} for 0 j < M ad lt A {A 0,A 1,...,A M 1 } b a parttotyp (g,{k 0,...,K M 1 }, g s )-BNCDP of sz l ovr Z g wth d A s for 1 g s, whr A j {A j r : 0 r < l}. For 0 j < M, st A j {maj r : 0 r < l}, B j (k, ) Bj k + sm for 0 k < u ad 0 < g s, ad D j {B j (k,) : 0 k < u,0 < g s } A j, th th sz of D j s gu s +l. t D {D j : 0 j < M}. It rmas to prov that D s a partto-typ (mg,{k 0 K 0,...,K M 1 K M 1 }, g s )- BNCDP ovr Z mg wth d D sm for 1 g s. Sc all lmts of bas blocks of B j, togthr wth 0,m,...,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg, w hav {B j (k,) : 0 k < u,0 < g s } s a partto of Z mg \ mz mg. Sc A j s a partto-typ (g,{k 0,...,K M 1 }, g s )-BNCDP, w hav that A j s a partto of mz mg ad D j s a partto of Z mg. For 0 j j < M, E (D j,d j ) ( ) g u 1 s 1 E (B j (k, ),Bj (k, ) ) E (A j,a j ) k0 0 ( g s 1 0 E (B j + sm,b j + sm)) E (A j,a j ), g s [ E(B j,b j )] (m E (A j,a j )). Sc {B j : 0 j < M} s a (mg,g,{k 0,...,K M 1 },1)- BNCRDP, w hav that (B j ) Z mg \ mz mg ad E (B j,b j ) Z mg \ mz mg for 0 j j < M. Sc {A j : 0 j M 1} s a (g,{k 0,...,K M 1 }, g s )- BNCDP, w hav that (A j ) m (A j) cotas ach ozro lmt of mz mg at most g s tms ad E(A j,a j ) m (A j,a j ) cotas ach lmt ofmz mg at most g s tms for j j. It follows that E (D j,d j ) cotas ach lmt of Z mg at most g s tms. Smlarly, (D j) cotas ach ozro lmt of Z mg at most g s tms. Hc, D s a parttotyp (mg,{k 0 K 0,...,K M 1 K M 1 }, g s )-BNCDP of sz gu s +l ovr Z mg. From th costructo, t s asy to s that f τ E (B j,b j ), th th orbt cycl D (j,j )(τ) of τ (D j,d j ) s of th form (a 0,a 0 +sm,a 0 +2sm,...,a 0 +mg sm), ad f τ E (A j,a j ) th th orbt cycl D (j,j )(τ) of τ (D j,d j ) s of th form (mb 0,mb 1,...,mb s 1) whr (b 0,b 1,...,b s 1) s th orbt cycl of τ m (A j,a j ), othrws D (j,j )(τ). Th d (Dj,D j ) (τ) sm for τ (B j,b j ), d (Dj,D j ) (τ) sm for τ E (A j,a j ) bcaus d(aj,a j ) s for 1 g s ad d (Dj,D j ) (τ) mg for τ / E (D j,d j ). It follows that d (Dj,D j ) m{d (Dj,D j ) (τ) : 1 τ < mg} sm for 1 g s. Smlarly, t s radly chckd that d Dj sm for 1 g s. Thrfor, d D j ) m{ m {d(dj,d 0 j j }, m <M 0 j<m {ddj }} sm for 1 g s, ad D s th rqurd BNCDP. Ths complts th proof. IV. COMBINATORIA CONSTRUCTIONS A. Drct costructos of strctly optmal (,2; 2 )-FHSs Costructo A1: t u b a postv tgr ad lt X {x(t)} t0 2u 1 b th FHS of lgth 2u ovr Z u dfd by x(t) x(2t 1 + t 0 ) ( 1) t0 t 1, whr t 2t 1 + t 0, 0 t 0 < 2, 0 t 1 < u. Thorm 4.1: For a postv tgr u, th FHS X Costructo A1 s a strctly optmal (2u, 2; u)-fhs, ad H(X;) u for 1 2u. Proof: For 0 r < u, dot B r {t : x(t) r,0 t < 2u}, th w hav B j {2j,1 2j} (mod 2u) for 0 j < u, D(1 4j) (2j,2j +u) (mod 2u) for 0 j u 1 2, D(1 4j) (2j u,2j) (mod 2u) for u+1 2 j < u, D(k), f k ±(1 4j), 0 j < u.

8 8 By th fact m{g : g {2u} ( D(τ))} m{g : g {2u} ( D( τ))}, w hav d 1 u ad d 2 2u. Th, {B 0,,B u 1 } s a partto-typ (2u,2,2)-CDP of sz u 2u wth d u for 0 < 2. Thrfor, by Thorm 2u u.5, X s strctly optmal wth rspct to th boud (2). Rmark: Wh u s a odd tgr, a strctly optmal (2u,2;u)-FHS ca b also obtad from [6]. Costructo A2: t b a odd tgr ad lt X {x(t)} t0 1 b th FHS of lgth ovr Z ( 1)/2 dfd by x(t) j for t B j, 0 j < 1 2, whr B j ar dfd as follows: If 8a+1, th B 0 {0,4a+1,8a}; B 1 {4a 1,4a}; B 1+r {r,2a 2+2r} for 1 r a; B a+1+r {a+r,2a 1+2r} for 1 r a 1; B 2a+r B 1+r +4a+1 (mod ) for 1 r 2a 1. If 8a+, th B 0 {0,4a+1,4a+2}; B 1 {2a,6a+}; B 2 {2a+1,6a+1}; B {6a+2,6a+4}; B +r {r,4a+1 r} for 1 r 2a 1; B 2a+2+r B +r +4a+2 (mod ) for 1 r 2a 2. If 8a+5, th B 0 {0,4a+2,4a+}; B 1 {2a+1,6a+5}; B 2 {2a+2,6a+}; B {1,6a+4}; B +r {2a+2+r,6a+ r} for 1 r 2a 1; B 2a+2+r B +r +4a+ (mod ) for 1 r 2a 1. If 8a+7, th B 0 {0,4a+,4a+4}; B r {r,2a+2r} for 1 r a+1; B a+r {a+1+r,2a+1+2r} for 1 r a; B 2a+r B r +4a+4 (mod ) for 1 r 2a+1. Thorm 4.2: For ay odd tgr 5, th FHS X Costructo A2 s a strctly optmal (,2; 1 2 )-FHS ad H(X;) 2 for 1. Proof: By Thorm.5, w oly d to show that {B j : 0 j < 1 2 } s a partto-typ (,{2,},2)-CDP wth d ( 1)/2 2 for 1 ( 1)/2 2. Clarly, B {B j : 0 j < 1 2 } s a partto of Z. By th fact m{g : g {} ( D(τ))} m{g : g {} ( D( τ))}, w oly d to comput D(τ) for 1 τ 1 2, so as to obta d. Accordg to th costructo, w dvd t to four cass. Cas 1: 8a+1. By Costructo A2, w hav that D(1) (4a 1,8a), D(4a) (1+4a), D(4a 1) (1+4a), D(2a 2+r) (r,r +4a+1),1 r a, ad D(a 1+r) (a+r,5a+1+r),1 r a 1. Th, t holds that d 1 4a ad max{ D(τ) : 0 < τ < 4a} 2. Thrfor, by Thorm.5, th squc X s strctly optmal wth rspct to th boud (2). Cas 2: 8a+. By Costructo A2, w hav D(1) (4a+1), D(4a+1) (0,2+4a), D(2) (2+6a), D(4a) (1+2a,+6a), D() (2a 1), ad D(4a+1 2r) (r,r+4a+2),1 r 2a 2. Th, t holds that d 1 4a+1 ad max{ D(τ) : 0 < τ < 4a + 1} 2. Thrfor, by Thorm.5, th squc X s strctly optmal wth rspct to th boud (2). Cas : 8a+5. By Costructo A2, w hav D(1) (4a+2), D(4a+2) (0,+4a), D(2a+2) (4+6a), D(4a+1) (2a+2,5+6a), ad D(4a+1 2r) (2a+2+r,r+6a+5),1 r 2a 1. Th, t holds that d 1 4t+2 ad max{ D(τ) : 0 < τ < 4a + 2} 2. Thrfor, by Thorm.5, th squc X s strctly optmal wth rspct to th boud (2). Cas 4: 8t+7. By Costructo A2, w hav D(1) (4t+), D(4t+) (0,4+4t), D(2t+r) (r,r+4t+4) for 1 r t+1, ad D(t+r) (t+1+r,r+5t+5), 1 r t. Th, t holds that d 1 4a+ ad max{ D(τ) : 0 < τ < 4t+} 2. Thrfor, by Thorm.5, th squc X s strctly optmal wth rspct to th boud (2). Ths complts th proof. B. Strctly optmal FHSs from rsolvabl CRDFs I ths subscto, w costruct rsolvabl CRDFs by usg cyclotomc classs ordr to obta strctly optmal FHSs. Rsolvabl CRDFs hav b tsvly studd dsg thory, s [12]. W quot som rsults of rsolvabl CRDFs th followg lmma. mma 4.: (1) Thr xsts a rsolvabl (v,2,,1)-crdf for all v of th form 2 k p 1 p 2 p s whr k {1,5} ad ach p j 1 (mod 12) s a prm ([], [4]). (2) Thr xsts a rsolvabl (v,2,,1)-crdf for all v of th form 8p 1 p 2 p s whr ach p j 1 (mod 6) s a prm ([]). () Thr xsts a rsolvabl (v,,4,1)-crdf for all v of th form p 1 p 2 p s whr ach p j 1 (mod 4) s a prm ([4]). Th applcato of mma.6 to th CRDFs mma 4. ylds th followg strctly optmal FHSs. Thorm 4.4: (1) Thr xsts a strctly optmal(v,2; v+1 )- FHS for all v of th form 2 k p 1 p 2 p s whr k {1,5} ad ach p 1 (mod 12) s a prm. (2) Thr xsts a strctly optmal (v,2; v+1 )-FHS for all v of th form 8p 1 p 2 p s whr ach p 1 (mod 6) s a prm.

9 9 () Thr xsts a strctly optmal (v,; v+1 4 )-FHS for all v of th form p 1 p 2 p s whr ach p 1 (mod 4) s a prm. t q b a prm powr wth q f + 1 ad GF(q) b th ft fld of q lmts. Gv a prmtv lmt α of GF(q), df C0 {αj : 0 j f 1}, th multplcatv group gratd by α, ad C α C 0 for 1 1. Th C0,C 1,...,C 1 partto GF(q) GF(q) \{0}. Th C (0 < ) ar kow as cyclotomc classs of dx (wth rspct to GF(q)). Gv a lst {a 1,a 2,...,a } of lmts GF(q), f ach cyclotomc class C, 0, cotas xactly o lmt of th lst, th w say that th lst forms a complt systm of dstct rprstatv of cyclotomc classs. I th thory of cyclotomy, th umbrs of solutos of x+1 y, x C, y C j ar calld cyclotomc umbrs of ordr rspct to GF(q) ad dotd by (,j). mma 4.5: Thr xsts a rsolvabl (4p, 4,, 1)-CRDF for ay prm p 7 (mod 12). Proof: t ε b a prmtv sxth root of uty Z p. Clarly, ε 5 s also a prmtv sxth root of uty ad 1 ε + ε 2 ε 0. Thus, 5 +1 ε+1 ε2 +1 ε+1 1 ε ε 2 C1 2 sc 1 C1 2. Wthout loss of gralty, w may assum that ε + 1 C0 2. Sc gcd(4,p) 1, w hav that Z 4p s somorphc to Z 4 Z p. Dot by R a complt systm of rprstatvs for th costs of {1,ε 2,ε 4 } C0 2, ths mpls that {rε 2j : r R,0 j < } C0 2. (5) St Th B {{(0,r),(0,rε 2 ),(0,rε 4 )} : r R} {{(1, rε 2 ),(2, r),(, rε)} : r R} B B {{(1, rε 4 ),(2, rε 2 ),(, rε )} : r R} {{(1, r),(2, rε 4 ),(, rε 5 )} : r R}. B ( {1} { r, rε, rε 2, rε, rε 4, rε 5 }) ( {0} {r,rε 2,rε 4, r, rε 2, rε 4 }) (mod {(0,0),(2,0)}). I vw of (5), s ad 1,ε C2 1, w hav B {(0,0),(1,0)} B B 1 {} Z p (mod {(0,0),(2,0)}). 0 Hc, all lmts of bas blocks of B, togthr wth (0,0) ad (1, 0), form a complt systm of rprstatvs for th costs {(0,0),(2,0)} Z 4 Z p. It s lft to chck that B s a (4p,4,,1)-CRDF. It s straghtforward that (B) {z} z, z0 whr j0 j0 1. j0 2 {±rε 2j (1 ε 2 )}, 2 {rε 2j (ε 2 1),rε 2j (1 ε)}, 2 {±rε 2j (ε 2 ε)}, I vw of (5), 1 C 2 1 ad ε+1 C 2 0, th w hav: (B) Z 4 (Z p \{0}). Thrfor, B s a rsolvabl (4p,4,,1)-CRDF. mma 4.6: Thr xsts a rsolvabl (6p, 6,, 1)-CRDF for ay prm p 5 (mod 8). Proof: For p 5, t xsts by Exampl.4. For othr prm p 5 (mod 8), lt ε b a prmtv fourth root of uty Z p, th ε C1 2. t ε C4 a ad 1 + ε C4 2 j, whr a {1,} ad j {0,1,2,}. W frst show that thr 2ε 1+ε (z +1), 2ε xsts a lmt z such that 1+ε (zε) C4 2. For ths purpos, w d th cyclotomc umbrs (j a,j) 4. For ay prm p 5 (mod 8), thy ar gv pag 48 of [27] (0,1) 4 (,2) 4 p+1+2x 8y 16, (0,) 4 (1,2) 4 p+1+2x+8y 16, ad (1,0) 4 (2,1) 4 (2,) 4 (,0) 4 p 2x 16, whr p x 2 +4y 2 wth x 1 (mod 4). Sc p x 2 +4y 2, w hav that x p ad (x ± 4y) 2 5(x 2 + 4y 2 ) 5p. Thus p+1+2x±8y 16 p+1 2 5p 16 > 0 ad p 2x 16 p 2 p 16 > 0 for p 29,.., for ay 0 j < 4, th cyclotomc umbr (j a,j) 4 > 0 f p 29. Hc, thr xsts a lmt z j Cj a 4 such that z j+1 Cj 4. Thus, for ach p 5 (mod 8) wth p 29 thr xsts a lmt z 2ε such that 1+ε (z +1), 2ε 1+ε (zε) C4 2. For p 1, tak ε 8 ad z 11. Th 2ε 1+ε (z +1), 2ε 1+ε (zε) C4 2. Sc gcd(6,p) 1, w hav that Z 6p s somorphc to Z 6 Z p. St B {{(0,w),(0, w),(1,wε)} : w C 4 0 } {{(1, w),(,wε),(, wε)} : w C 4 0 } {{(2, wε(2z +1)),(4, wε),(5,wε(2z +1))} : w C 4 0 } Th B B {{(2,w(2z +1)),(4,w),(5, w(2z +1))} : w C 4 0 }. B w C 4 0 ( w C 4 0 {0,1} {w, w,wε, wε} {2} {w(2z +1), w(2z +1), w(2z +1)ε, w(2z +1)ε}) (mod {(0,0),(,0)}).

10 10 Sc ε C1 2 ad 1 C4 2, w hav ( B) {(0,0),(1,0),(2,0)} B B 2 {} Z p (mod {(0,0),(,0)}). 0 Hc, all lmts of bas blocks of B, togthr wth (0,0),(1,0) ad (2,0), form a complt systm of rprstatvs for th costs {(0,0),(,0)} Z 6 Z p. It s lft to chck that B s a (6p,6,,1)-CRDF. It s straghtforward that whr 0 1 w C w C 4 0 w C 4 0 (B) 5 {b} b, w C 4 0 b0 {±2w, ±2wε}, {w(ε 1),w(1+ε),wε(2z +2), w(2z +2)}, {w(1+ε),w(1 ε),2wzε, 2wz)}, {±2wε(2z +1),±2w(2z +1)}, 4 2, ad 5 1. Sc ε C1 2, 1 C4 (2z+2)ε 2, ε(1+ε) ε 1 ad 1+ε, 2z 1+ε C2 4. W hav that (B) Z6 (Zp \{0}). Thrfor, B s a rsolvabl (6p,6,,1)-CRDF. Th applcato to th CRDFs ths subscto ylds th followg strctly optmal FHSs. Corollary 4.7: (1) Thr xsts a strctly optmal (4p,2; 4p+2 )-FHS for ay prm p 7 (mod 12). (2) Thr xsts a strctly optmal (6p,2;2p+ 1)-FHS for ay prm p 5 (mod 8). Proof: I th followg, w oly prov th cas (1). Th othr cas ca b hadld smlarly. By mma 4.5, thr xsts a rsolvabl (4p,4,,1)-CRDF for ay prm p 7 (mod 12). By Thorm 4.1 ad Thorm.5, thr s a partto-typ (4,2,2)-CDP of sz 2 ovr Z 4 wth d 2 for 1 2. By applyg mma.6 wth s 2, w obta a partto-typ (4p,{2,},2)-CDP ovr Z 4p wth d 2p for 1 2. Furthr, applyg Thorm.5 ylds a strctly optmal (4p,2; 4p+2 )-FHS. C. A cyclotomc costructo of strctly optmal FHS sts I ths subscto, w obta a class of partto-typ BNCDPs wth a spcal proprty by usg cyclotomc classs, from whch w obta a w costructo of strctly optmal FHS sts. t v b a odd tgr wth v > 1 ad dot by U(Z v ) th st of all uts Z v. A lmt g U(Z v ) s calld a prmtv root modulo v f ts multplcatv ordr modulo v s ϕ(v), whr ϕ(v) dots th Eulr fucto whch couts th umbr of postv tgrs lss tha ad coprm to v. It s wll kow that for a odd prm p, thr xsts a lmt g such that g s a prmtv root modulo p b for all b 1 [1]. t v b a odd tgr of th form v p m1 1 pm2 2 p ms s for s postv tgrs m 1,m 2,...,m s ad s dstct prms p 1,p 2,...,p s. t b a commo factor of p 1 1,p 2 1,...,p s 1 ad > 1. Df f m{ p 1 : 1 s}. For 1 s, lt g b a prmtv root modulo p m. By th Chs Rmadr Thorm, thr xst uqu lmts g,a U(Z v ) such that g g fpm 1 (mod p m ) for all 1 s, a g (mod p m ) for all 1 s, th th multplcatv ordr of g modulo v s, th lst of dffrcs arsg from G {1,g,...,g 1 } s a subst of U(Z v ) ad a t g c g c U(Z v ) for 1 t < f ad 0 c,c <. mma 4.8: t v b a postv tgr of th form v p m1 1 pm2 2 p ms s for s postv tgrs m 1,m 2,...,m s ad s dstct prms p 1,p 2,...,p s. t u b a postv tgr such that gcd(u,v) 1. t b a commo factor of u,p 1 1,p 2 1,...,p s 1 ad > 1, ad lt f m{ p 1 : 1 s}. Th thr xsts a (uv,u,{k 0,...,K f 1 },1)-BNCRDP of sz v 1 such that all lmts of bas blocks of ach CRDP, togthr wth 0, form a complt systm of rprstatvs for th costs of vz uv Z uv whr K 0 K f 1 {}. Proof: t g ad a b dfd as abov ad dot G {1,g,...,g 1 }. Th G s a multplcatv cyclc subgroup of ordr of Z v ad (G) U(Z v ). For x,y Z v \ {0}, th bary rlato dfd by x y f ad oly f thr xsts a g G such that xg y s a quvalc rlato ovr Z v \ {0}. Th ts quvalc classs ar th substs xg,x Z v \ {0}, of Z v. Dot by R a systm of dstct rprstatvs for th quvalc classs modulo G of Z v \ {0}, th R v 1 ad {ra b g j : r R, 0 j < } Z v \{0}, (6) for ay tgr b. Sc gcd(u,v) 1, w hav that Z uv s somorphc to Z u Z v. t B b r {(ju,rab g j ) : 0 j < } for r R ad 0 b < f, B b {B b r : r R}. W clam that {B b : 0 b < f} s a (uv,u,{{},{},...,{}},1)-bncrdp such that all lmts of bas blocks of B t, togthr wth (0,0), form a complt systm of rprstatvs for th costs of Z u {0} Z u Z v. I vw of qualty (6), t holds that Br b {(0,0)} {( ju,rab g j ) : 0 j < } {(0,0)} {0} Z v (mod Z u {0}). It follows that all lmts of bas blocks of B t, togthr wth (0,0), form a complt systm of rprstatvs for th costs of Z u {0} Z u Z v. For 0 b < f, w show that (B b ) cota ach o-zro lmt of Z u Z v at most oc.

11 11 I vw of qualty (6), w gt (B b ) ({( ju,rab g j ) : 0 j < }) {( j u,rab g j ) ( ju,rab g j ) : 0 j j < } {( (j j)u,ra b (g j g j )) : 0 j j < } {( cu,rab g j (g c 1)) : 0 j <,1 c < } ( u Z u \{0}) (Z v \{0}). Hc, ach B b s a (uv,u,,1)-crdp. For 0 b b < f, accordg to qualty (6) ad a b b g c 1 U(Z v ), w gt (B b,b b ) (Br,B b r b ) {( ku,rab g k ) ( ju,rab g j ) : 0 j,k < } {( (k j)u,r(a b g k a b g j )) : 0 j,k < } {( cu,rab g j (a b b g c 1)) : 0 j <,0 c < } ( u Z u) (Z v \{0}). Hc, (B b,b b ) cotas ach lmt of Z u Z v at most oc. Ths complts th proof. Start wth a (uv,u,{k 0,...,K f 1 },1)-BNCRDP mma 4.8 whr K 0 K f 1 {}. t A k { u j + k : 0 j < } ad A b {A k : 0 k < u } for 0 b < f, t s asy to s that {A 0,...,A f 1 } s a partto-typ (u,{k 0,...,K f 1 },u)-bncdp of sz u wth d for 1 u. Applyg mma.11 wth g u ad s 1 ylds th followg corollary. Corollary 4.9: Suppos that th paramtrs v,, f, u ar th sam as thos th hypothss of mma 4.8. Th thr xsts a (uv,{k 0,...,K f 1 },u)-bncdp of sz uv wth d v for 1 u whr K 0 K f 1 {}. Furthrmor, wh u, by Thorm.10 th BNCDP Corollary 4.9 s a strctly optmal FHS st. So, w hav th followg corollary. Corollary 4.10: ([6]) Suppos that th paramtrs v,, f ar th sam as thos th hypothss of mma 4.8. Th thr xsts a strctly optmal (v, f, ; v)-fhs st S wth partal Hammg H(S;) v for 1 v. mma 4.8 trprts th gralzd cyclotomc costructo [6] for FHS sts va cyclotomc costss. I comparso, our mthod s qut at ad mor clar to udrstad. V. A DIRECT CONSTRUCTION OF STRICTY OPTIMA FHS SETS I ths scto, w us ft flds to gv a drct costructo of strctly optmal FHS sts. Costructo B t p b a prm ad lt m b a tgr wth m > 1. t α b a prmtv lmt of GF(p m ) ad dot R { m 1 1 a α : a GF(p),1 < m}. t S {X a : a R} b a st of p m 1 FHSs of lgth p(p m 1), whr X a {X a (t)} p(pm 1) 1 t0 s dfd by X a (t) α t p m 1 + t p + a ad z u dots th last ogatv rsdu of z modulo u for ay postv tgr u ad ay tgr z. Thorm 5.1: t p b a prm prm ad lt m b a tgr wth m > 1. Th th FHS st Costructo B s a strctly optmal (p(p m 1),p m 1,p;p m )-FHS st wth rspct to th boud (4) ovr th alphabt GF(p m ). Proof: Frstly, w prov that H(S;) p m 1 for 1 p(p m 1). For 0 τ,j p(p m 1) 1 ad a,b R, th partal Hammg corrlato H X a,xb(τ) s gv by H X a,x j+ 1 tj j+ 1 tj j+ 1 b(τ : j ) tj h[x a (t),x b (t+τ)] h[α t p m 1 + t p +a,α t+τ p m 1 + t+τ p +b] h[a b τ p,α t p m 1 (α τ p m 1 1)]. Dot τ 0 τ pm 1 ad τ 1 τ p. Accordg to th valus of a,b ad τ, w dstgush four cass. Cas 1: a b ad τ 1 0. I ths cas τ 0 0 ad α τ0 1 GF(p m )\{0}. Th H Xa,X j+ 1 a(τ : j ) 0. tj h[0,α t p 1 (α τ 0 1)] Cas 2: a b ad τ 1 0. If τ 0 0, th H Xa,X j+ 1 a(τ : j ) tj h[ τ 1,0] 0. Othrws, lt t 0 b a tgr such that τ 1 α t0 p m 1 (α τ 0 1). Th H Xa,X j+ 1 a(τ : j ) tj j+ 1 tj p m 1 h[ τ 1,α t p m 1 (α τ 0 1)] {t : t t 0 (mod p m 1)}. Cas : a b ad τ 0 0. Sc a b / GF(p), w hav a b τ 1. Th j+ 1 H X a,x b(τ : j ) h[a b τ 1,0] 0. tj

12 12 Cas 4: a b ad τ 0 0. Clarly, a b τ 1 0. t t 0 b a tgr such that a b τ 1 α t0 p m 1 (α τ 0 1). Th, H X a,xb(τ : j ) j+ 1 tj j+ 1 tj p m 1 h[a b τ 1,α t p m 1 (α τ 0 1)] {t : t t 0 (mod p m 1)}. I summary, th dscusso th four cass abov shows that H(S;) p m 1. Fally, w prov that H(S; ). Not that S p m 1 cotas p m 1 FHSs of lgth p(p m 1) ovr a alphabt of sz p m. Sc I p(pm 1)p m 1 p p m 1, by mma 2.4 m w gt H(S; ) p(p m 1) p(p m 1) p m 1. 2(p m 1)p(p m 1)p m 1 p m (p m 1)p m (p(p m 1)p m 1 1)p m 1 p p(2pm ) (p m 1)p m 1 Thrfor, {X a : a R} s a strctly optmal (p(p m 1),p m 1,p;p m )-FHS st wth rspct to th boud (4). W llustrat th da of Thorm 5.1 wth m 2 adp th followg xampl. Exampl 5.2: Usg th prmtv polyomal f(x) x 2 + x+2 GF()[x], w costruct th GF(9) as GF()[α]/f(α) whr α 2 + α Th 9 lmts of GF(9) ca b rprstd th form a 0 +a 1 α,a 0,a 1 GF(). Th FHS st S {X 0,X α,x 2α } gratd by Costructo B s gv by X 0 (1,α+1,2α,2α+2,0,2α+2,α+2, α+2,0,α,2α+2,2α+1,2,2α+1,α+1, α+1,2,α+2,2α+1,2α,1,2α,α,α) X α (α+1,2α+1,0,2,α,2,2α+2, 2α+2,α,2α,2,1,α+2,1,2α+1, 2α+1,α+2,2α+2,1,0,α+1,0,2α,2α) X 2α (2α+1,1,α,α+2,2α,α+2,2, 2,2α,0,α+2,α+1,2α+2,α+1,1, 1,2α+2,2,α+1,α,2α+1,α,0,0). It s radly chckd that S {X 0,X α,x 2α } s a strctly optmal (24,,;9)-FHS st wth rspct to th boud (4). Such a FHS st s th sam as that Exampl.9. VI. TWO RECURSIVE CONSTRUCTIONS OF STRICTY OPTIMA FHS SETS I ths scto, two rcursv costructos ar usd to costruct strctly optmal dvdual FHSs ad FHS sts. Th frst rcursv costructo s basd o th cyclc dffrc matrx (CDM). A (w,t,1)-cdm s a t w matrxd (d j ) (0 t 1, 0 j w 1) wth trs from Z w such that, for ay two dstct rows R r ad R h, th vctor dffrc R h R r cotas vry rsdu of Z w xactly oc. It s asy to s that th proprty of a dffrc matrx s prsrvd v f w add ay lmt of Z w to all trs ay row or colum of th dffrc matrx. Th, wthout loss of gralty, w ca assum that all trs th frst row ar zro. Such a dffrc matrx s sad to b ormalzd. Th (w,t 1,1)- CDM obtad from a ormalzd (w, t, 1)-CDM by dltg th frst row s sad to b homogous. Th xstc of a homogous (w,t 1,1)-CDM s quvalt to that of a (w, t, 1)-CDM. Obsrv that dffrc matrcs hav b xtsvly studd. A larg umbr of kow (w, t, 1)-CDMs ar wll documtd [12]. I partcular, th multplcato tabl of th prm fld Z p s a (p,p,1)-cdm. By usg th usual product costructo of CDMs, w hav th followg xstc rsult. mma 6.1: ([12]) t w ad t b tgrs wth w t. If w s odd ad th last prm factor of w s ot lss tha t, th thr xsts a (w,t,1)-cdm. Thorm 6.2: Assum that {B 0,...,B M 1 } s a (mg,g,{k 0,K 1,...,K M 1 },1)-BNCRDP of sz u such that all lmts of bas blocks of B j, togthr wth 0,m,...,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg for 0 j < M, whr s g ad B j {B j 0,Bj 1,...,Bj u 1 }. If thr xsts a homogous (w,t,1)-cdm ovr Z w wth t max { M B j 0 k<u k } ad gcd(w, g s ) 1, th thr also j1 xsts a (mgw,gw,{k 0,K 1,...,K M 1 },1)-BNCRDP of sz uw, B {B 0,...,B M 1 } such that all lmts of bas blocks of B j, togthr wth 0,m,...,(sw 1)m, form a complt systm of rprstatvs for th costs of smwz mgw Z mgw for 0 j < M. Proof: t Γ (γ,j ) b a homogous (w,t,1)-cdm ovr Z w. For ach collcto of th followg M blocks: B 0 {a,0,1,...,a,0,k0 }, B 1 {a,1,k 0+1,...,a,1,k1 },. B M 1 {a,m 1,kM 2+1,...,a,M 1,kM 1 }, whr 0 < u, w costruct th followg uw w blocks: B j (,k) {a,j,k j 1+1 +mgγ kj 1+1,k,...,a,j,kj +mgγ kj,k}, St 0 j < M,0 k < w. B j {Bj (,k) : 0 < u,0 k < w},ad B {B j : 0 j < M}, th th sz of B j s uw for 0 j < M. It s lft to show that B th rqurd (mgw,gw,{k 0,K 1,...,K M 1 },1)- BNCRDP.

13 1 Frstly, w show that all lmts of bas blocks B j, togthr wth 0,m,...,(sw 1)m, form a complt systm of rprstatvs for th costs of smwz mgw Z mgw. Sc gcd(w, g g s ) 1, w hav that {c s : 0 c < w} {0,1,...,w 1} (mod w). It follows that {ms cg s : 0 c < w} {csm : 0 c < w} (mod smw). Clarly, 0 k<w Bj (,k) z B{z j + cmg : 0 c < w}. Sc all lmts of bas blocks of B j, togthr wth 0,m,2m,,(s 1)m, form a complt systm of rprstatvs for th costs of smz mg Z mg, w hav that 0 <u Bj {0,1,2,...,sm 1} \ {0,m,...,sm m} (mod sm) ad B j (,k) 0 <u 0 k<w 0 <uz B j 0 <uz B j z I sm\{0,m,...,sm m} {z +cmg : 0 c < w} {z +csm : 0 c < w} {z +csm : 0 c < w} I sw \{0,m,...,smw m} (mod smw), as dsrd. Scodly, w show that ach B j s a (mgw,gw,k j,1)- CRDP. ScB j s a(mg,g,k j,1)-crdp, w hav (B j ) Z mg \mz mg. Smpl computato shows that (B j ) (B j (,k) ) 0 <u, 0 k<w {a b+cmg : a b B j, 0 c < w} 0 <u (mgz mgw +τ) Z mgw \mz mgw. τ (B j) It follows that B j s a (mgw,gw,k j,1)-crdp. Fally, w show (B j,b j ) Z mgw \ mz mgw for 0 j j < M. Sc (B j,b j ) Z mg \mz mg, w gt (B j,b j ) 0 <u 0 k<w (B j (,k),bj (,k) ) {b a+cmg : (a,b) B j Bj, 0 c < w} 0 <u (mgz mgw +τ) Z mgw \mz mgw. τ (B j,b j ) Thrfor, B s th rqurd BNCRDP. Ths complts th proof. Th sgfcac of Thorm 6.2 s that t gvs us a ffctv way to costruct a BNCRDP such that all lmts of bas blocks of ach CRDP, togthr wth 0, m,...,(sw 1)m, form a complt systm of rprstatvs for th costs of smwz mgw Z mgw, whch s crucal to our vstgato of ths papr. By mployg Thorm 6.2, w obta th followg costructo for rsolvabl CRDFs. Corollary 6.: Assum that thr xsts a rsolvabl (mg, g, k, 1)-CRDF, a rsolvabl (gw, g, k, 1)-CRDF ad a (w,k + 1,1)-CDM. If gcd(w,k 1) 1, th thr xsts a rsolvabl (mgw, g, k, 1)-CRDF. Proof: Sc thr xsts a (w, k + 1, 1)-CDM, thr xsts a homogous (w, k, 1)-CDM. Sc thr xsts a rsolvabl (mg,g,k,1)-crdf ad gcd(w,k 1) 1, applyg Thorm 6.2 wth M 1 ad s g k 1 gvs a rsolvabl (mgw,gw,k,1)-crdf B. t A b a rsolvabl (gw,g,k,1)-crdf. Th, B {ma : A A} s a rsolvabl (mgw,g,k,1)-crdf, whr ma {ma : a A}. Rmark: Wh g k 1, th rcursv costructo for CRDFs from Corollary 6. has b gv [2]. Applyg Corollary 6. wth th kow rsolvabl CRDFs scto IV-B, w gt th followg w rsolvabl CRDFs. Corollary 6.4: (1) Thr xsts a rsolvabl (v, 4,, 1)- CRDF for all v of th form 4p 1 p 2 p u whr ach p j 7 (mod 12) s a prm. (2) Thr xsts a rsolvabl (v,6,,1)-crdf for all v of th form 6p 1 p 2 p u whr ach p j 5 (mod 8) s a prm. Proof: W oly prov th frst cas. Th othr cas ca b hadld smlarly. W prov t by ducto o t. For u 1, by mma 4.5 th assrto holds. Assum that th assrto holds for u r ad cosdr u r + 1. Start wth a rsolvabl (4p 1 p 2 p r,4,,1)-crdf ovr Z 4p1p 2 p r whch xsts by ducto hypothss. By mma 6.1 ad mma 4.5, thr xsts a (p r+1,5,1)-cdm ad a rsolvabl (4p r+1,4,,1)-crdf. Applyg Corollary 6. ylds a rsolvabl (4p 1 p 2 p r+1,4,,1)-crdf. So, th cocluso holds by ducto. Th applcato to th CRDFs Corollary 6.4 gvs th followg strctly optmal FHSs. Thorm 6.5: (1) Thr xsts a strctly optmal(v,2; v+2 )- FHS for all v of form 4p 1 p 2...p u whr ach p j s a prm 7 (mod 12). (2) Thr xst a strctly optmal (v,2; v +1)-FHS for all v of form 6p 1 p 2...p u whr ach p j 5 (mod 8) s a prm. Proof: W oly prov th scod cas. Th frst cas ca b hadld smlarly. By Corollary 6.4, thr xsts a rsolvabl (v,6,,1)-crdf. By Thorm 4.1 ad Thorm.5, thr xsts a (6,2,2)-CDP, A of sz wth d A 6 2 for 1 2. Applyg mma.6 wth g 6 ad s, thr xsts a partto-typ (v,{2, }, 2)-CDP ovr Z v wth d v 2 for 1 2. Applyg Thorm.5 wth l + v 6 v+ v+, w obta a strctly optmal (v,2; )- FHS. By vrtu of mma 4.8, w ca apply Thorm 6.2 to produc th followg srs of strctly optmal FHS sts mtg th lowr boud (4). Corollary 6.6: t v b a odd tgr of th form v p m1 1 pm2 2 p ms s for s postv tgrs m 1,m 2,...,m s ad s prms p 1,p 2,...,p s wth p 1 < p 2 < < p s. t b a commo factor of p 1 1,p 2 1,...,p s 1 such that 2 < 2 < p 1 ad lt f p1 1. t w b a odd tgr of th form w q 1 1 q2 2...qt t for t postv tgrs 1, 2,..., t ad t dstct prms q 1,q 2,...,q t wth q 1 <... < q t such that p 1 q 1 ad gcd(,w) 1. t r b a commo factor of,q 1 1,q 2 1,...,q t 1 wth r > 1. If v > ( 2), th thr xsts a strctly optmal (wv,f,;(v 1)w+ w r )-FHS st wth rspct to th boud (4). Proof: Frstly, w prov that thr xsts a partto-typ (wv,{k 0,...,K f 1 },)-BNCDP of sz (v 1)w+ w r, S

14 14 such that d S wv for 1, whr K 0 K f 1 {,r}. By mma 4.8, thr xsts a (v,,{k 0,...,K f 1 },1)- BNCRDP of sz v 1 such that all lmts of bas blocks of ach CRDP, togthr wth 0, form a complt systm of rprstatvs for th costs of vz v Z v whr K 0 K f 1 {}. Sc p 1 q 1, by mma 6.1 thr xsts a homogous (w,p 1 1,1)-CDM ovr Z w. Sc gcd(,w) 1, applyg Thorm 6.2 wth g,s 1 ylds a(wv,w,{k 0,K 1,...,K f 1 },1)-BNCRDP of sz (v 1)w such that all lmts of bas blocks of ach CRDP, togthr wth 0,v,...,(w 1)v, form a complt systm of rprstatvs for th costs of wvz wv Z wv. By Corollary 4.9, thr xsts a partto-typ (w,{k 0,...,K f 1 },)- BNCDP of sz w r, S wth ds w for 1 whr K 0 K f 1 {r}. By applyg mma.11 wth g w ad s w w obta a partto-typ (vw,{k 0,...,K f 1 },)-BNCDP of sz (v 1)w+ w r, S wth d S wv for 1, whr K 0 K f 1 {,r}. Fally, w show By dfto, w hav I If r, w hav 2IM (I+1)Il (M 1)M 2IM (I+1)Il (M 1)M vw p1 1 (v 1)w+ w r rv(p1 1) (v 1)r+. { p1 1 f r, ad p 1 2 f r <. 2(p1 1)wvf p 1(p 1 1)wv (wvf 1)f (p1 2)wv wv(p 1 1) 1 (wv 1) wv(p 1 1) 1. Othrws, w hav 2IM (I +1)Il (M 1)M 2(p1 2)wv p 1 1 (p 1 1)(p 1 2)(wv w+ w r ) (wv p 1 1 wv+ (p 1 2)w( r) r wv(p 1 1) 1 1) p 1 1. (7) Sc p 1 > 2, r 2 ad v > ( 2), w hav v(p 1 1) > v+ (p1 2)( r) r, whch lads to wv+ (p1 2)w( r) r 0 < < 1. wv(p 1 1) 1 It follows from (7) that 2IM (I+1)Il (M 1)M. By Thorm.10, S s a strctly optmal (wv,f,;(v 1)w+ w r )-FHS st wth rspct to th boud (4). Ths complts th proof. Now, w prst th scod rcursv costructo. Costructo C: t v b a odd tgr of th form v p m1 1 pm2 2 p ms s for s postv tgrs m 1,m 2,...,m s ad s dstct prms p 1,p 2,...,p s. t b a commo factor of p 1 1,p 2 1,...,p s 1 wth > 1, ad lt f m{ p 1 : 1 s}. t {X 0,X 1,...,X M 1 } b a (,M,;l)- FHS st ovr F, whr X {x (t)} t0 1 ad M f. For 0 < M, lt Y {y (t)} t0 v 1 b th FHS ovr Z v F, dfd by y (t) ( a g t t v,x ( t )) whr a ad g ar dfd Scto IV-C, z u dots th last ogatv rsdu of z modulo u for ay postv tgr u ad ay tgr z. mma 6.7: ty ady j b ay two FHSs Costructo C. Th for 0 τ,c < v ad 1 v, t holds that H Y,Y j (τ;c ) H X,X j ( τ ;k v) for som k wth 0 k <. Proof: By dfto, h[y (t),y j (t+τ)] h[( a g t t v,x ( t )),( a j g (t+τ) (t+τ) v,x j ( t+τ ))] h[ a g t t v, a j g (t+τ) (t+τ) v ] h[x ( t ),x j ( t+τ )] h[ (a a j g τ )t v, a j g τ τ v ] h[x ( t ),x j ( t+τ )]. Accordg to th paramtrs,j,τ, w dstgush two cass. Cas 1: j ad τ 0. Sc g τ 1 ad x ( t ) x j ( t+τ ), w gt H Y,Y j (τ;c ) c+ 1 c+ 1 tc tc h[0, a j g τ τ v] { f τ 0, 0 othrws. h[y (t),y j(t+τ)] Th last qualty holds sc gcd(v, ) 1. Thrfor, H Y,Y j (τ;c ) H X,X j ( τ ;k v) for 0 k <. Cas 2: j or τ 0. I ths cas, a a j g τ U(Z v ). St t 0 a j g τ τ (a a j g τ ) 1 (mod v). Th h[ (a a j g τ )t v, a j g τ τ v ] 1 f ad oly t v t 0 v. Sc (p r 1) for 0 < r s, t holds that v 1 (mod ). Th H Y,Y j (τ;c ) c+ 1 c+ 1 tc tc h[y (t),y j (t+τ)] h[ (a a j g τ )t v, a j g τ τ v] h[x ( t ),x j ( t+τ )] 0 a< t 0 +av {c,c+1,...,c+ 1} c+ 1 t 0 v a c t 0 v h[x ( t 0 +av ),x j ( t 0 +av +τ )] h[x ( t 0 +a ),x j ( t 0 +a+τ )] H X,X j ( τ ;t 0 + c t 0 v v ).