2000 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes

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1 2000 IEEE. Persoal use of this material is permitted. However, permissio to reprit/republish this material for advertisig or promotioal purposes or for creatig ew collective works for resale or redistributio to servers or lists, or to reuse ay copyrighted compoet of this work i other works must be obtaied from the IEEE.

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH [7] K. Iwata, M. Morii, ad T. Uyematsu, A efficiet uiversal codig algorithm for oiseless chael with symbols of uequal cost, IEICE Tras. Fudametals, vol. E80-A, o. 11, pp , Nov [8] R. M. Krause, Chaels which trasmit letters of uequal duratio, Iform. Cotrol, vol. 5, pp , [9] I. Csiszár ad J. Körer, Iformatio Theory, Codig Theorems for Discrete Memoryless Systems. New York: Academic, [10] T. S. Ha, Theorems o the variable-legth itrisic radomess, to be published. A New Recursive Uiversal Code of the Positive Itegers Hirosuke Yamamoto, Member, IEEE Abstract A ew recursive uiversal code of the positive itegers is proposed, i which ay give sequece ca be used as a delimiter of codeword while bit 0 is used as a delimiter i kow uiversal codes, e.g., Leveshtei code, Elias code, Eve Rodeh code, Stout code, Betley Yao code, etc. The codeword legth of the proposed code is shorter tha log i almost all of sufficietly large positive itegers although the kow codes are loger tha log for ay positive iteger. Idex Terms Elias code, log-star fuctio, uiversal code of positive itegers, uiversal codig. I. INTRODUCTION May researchers have treated the uiversal codig of the positive itegers that satisfy P () P ( +1); for ay 2N; (1) where P () is a probability distributio o the set of positive itegers N f1; 2; 3; 111g [1] [7]. These codes ca be used practically i various adaptive dictioary codes [8]. Besides the practical uses, it is a iterestig codig problem to cosider how efficietly we ca ecode the positive itegers uder the prefix coditio. Let log k 2 be the k-fold compositio of the fuctio ad let log 3 2 be log log logw () 2 (2) where w 3 () is the largest iteger w which satisfies log w 2 0. The, it is show theoretically that ay positive iteger ca be represeted with log w3 () bits if < e [2], [3]. O the other had, may researchers, e.g., Leveshtei [2], 1 Elias [4], Betley Yao [5], Eve Rodeh [6], Stout [7], etc., have proposed log 3 -type codes with a recursive structure to attai high performace i large. But, i their codes, codeword legth l() caot become shorter tha log2 3 although it satisfies l() log3 2 + w3 () +c where c is a costat. Mauscript received Jue 4, 1998; revised July 23, The material i this paper was preseted i part at the 1998 IEEE Iteratioal Symposium o Iformatio Theory, MIT, Cambridge, MA, August 16 21, The author is with the Departmet of Mathematical Egieerig ad Iformatio Physics, Uiversity of Tokyo, Hogo, Bukyo-Ku, Tokyo , Japa ( yamamoto@hy.t.u-tokyo.ac.jp). Commuicated by N. Merhav, Associate Editor for Source Codig. Publisher Item Idetifier S (00) I this correspodece, we propose a ew log 3 -type code with a recursive structure, which satisfies that l() log (1 0 20f )w 3 f () +c f eve i the worst cases ad l() log (1 + (1 0 20f ))w 3 f () +c f i the best cases. Here, f is a parameter of the code ad c f is a costat which depeds o f. wf 3 () is a similar fuctio to w3 (), which satisfies wf 3 () w 3 (). Sice the best ad worst cases occur at ifiitely may s, ad, roughly speakig, l() is distributed uiformly betwee two extreme cases, l() ca become shorter tha log2 3 i large parts of itegers. I Sectio II, we review Elias! code, which is a typical oe of the kow log 3 -type codes, ad we show the reaso why the codeword legth caot become shorter tha log2 3 i the kow codes. To overcome this defect, we devise a ew represetatio of biary umbers that ever has a give sequece as a prefix. I Sectio III, we propose a ew recursive uiversal code of the positive itegers based o the ew biary umber represetatio ad we evaluate the performace of the proposed code theoretically. It is show that the codeword legth of the proposed code is shorter tha log2 3 i almost all of sufficietly large positive itegers. The case of r-ary uiversal codes are treated i Sectio IV. We use the followig otatio i this correspodece. [] r is the ordiary r-ary umber of positive iteger such that the most sigificat digit of [] r is ozero. [] i r is the ordiary r-ary umber of with i digits. btc is the largest iteger ot exceedig t. Examples: [14] ; [14] ; [14] 3 112; [14] ; b 14c 3. II. NEW BINARY NUMBER REPRESENTATION EXCLUDING A FORBIDDEN PREFIX Elias! code C E () has the followig recursive structure [4]: C E( 0)[ K ] 2[ K01] [ 1] 2[ 0] 20 (3) where [] 2 is the ordiary biary umber of, the most sigificat bit (MSB) of which is always oe. Each k i (3) is determied recursively by k b k01 c. I other words, k +1represets the bit legth of [ k01 ] 2. The recursio i (3) stops whe the legth of [ K ] 2 is two. Fially, bit 0 is attached as a delimiter to idicate the ed of C E ( 0 ). 2 I the decodig, K is obtaied from the first two bits of C E ( 0 ), ad the legth of [ k01 ] 2 is recursively obtaied from k. Sice the MSB of every [ k ] 2 is 1, delimiter 0 ca stop the recursio ad [ 0 ] 2 ca easily be foud. Leveshtei W 2 code [2], Eve Rodeh code [6], ad Stout code [7] have similar structures ad their codes also use bit 0 as a delimiter i the same way as Elias! code. Leveshtei W 0 2 code [2] ad Betley Yao search-tree code [5] have a little differet structure. However, it is kow that their code ca be derived from Elias! like code by gatherig the MSB s of all [ k ] 2 ad delimiter 0 as a prefix. 1 Leveshtei code is the first log -type code although Elias! code is famous. 2 1 is the exceptio case, for which the codeword is defied as C (1) /00$ IEEE

3 718 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 We ote that the MSB of each [ k ] 2 is always 1. This meas that the MSB has o iformatio, or it is a redudat bit. But this redudat bit caot be omitted because the MSB is used to distiguish the delimiter 0. Sice each legth of [ k ] 2 is give by b k c +1 that is larger tha k, the codeword legth caot become shorter tha log 3 2 i the kow recursive codes of the positive itegers. The above ote suggests that if we use some sequece with legth f > 1, istead of 0, as a delimiter, the some of the redudat bits may be saved from a codeword. Whe 0 is used as a delimiter, the prefix, i.e., the MSB 1, of each [ k ] 2 does ot coicide with the delimiter 0. Hece, if we use a sequece [a] f 2, which is the ordiary biary umber of iteger a with f bits, as a delimiter, the we must devise a ew biary umber represetatio of the itegers, say B a; f (), such that the prefix of B a; f () does ot coicide with delimiter [a] f 2. Cosider biary sequeces whose legth is less tha j bits. The, the total umber of such biary sequeces is give by j01 2 j 0 2 while the umber of the biary sequeces with prefix [a] f 2 is give by j010f 2 j0f 0 1 if j 0 1 f or 0 if j f. This meas that the umber of biary sequeces ot havig prefix [a] f 2 is give by 2 j 0 2 (j0f) 0 1 for ay j 1 ad f 1, where (t) + is defied as TABLE I EXAMPLES OF B () AND ~B () (t) + maxft; 0g: (4) Hece, B a; f () ca be represeted by the followig formula: B a; f () where [ 0 M 2(j; f)] j 2 ; if M 2 (j; f) <M 2 (j; f) +N 2 (j; f; a); [ 0 M 2 (j; f 0 1)] j 2 ; if M 2(j; f) +N 2(j; f; a) <M 2(j +1;f) (5) M 2(j; f) 2 j 0 2 (j0f) (6) N 2 (j; f; a) b2 j0f ca 2j0f a; if j f 0; if j<f: Especially, if a 0, i.e., [a] f , the (5) ca be simplified as follows: B 0;f () [ 0 M 2 (j; f 0 1)] j 2 ; if M 2 (j; f) <M 2 (j +1;f): (8) We ote that lettig f () be the legth of B a; f (), the f () is give by f () j; if M 2 (j; f) <M 2 (j +1;f): (9) (7) where If [a] f 2 L(j; f) (f 0 1) 0 (f 0 j) + ; (11) ~N 2 (j; f; a) b2 j0f ac: (12) , the B ~ a; f () becomes ~B 0;f () [ 0 M 2(j; f 0 1) + L(j +1;f)] j 2 ; Some examples of B a; f () are show i Table I. Ay B a; f () does ot have [a] f 2 as a prefix. But, [a] f 2 may appear as a prefix whe B a; f () with legth f () <fis cocateated by aother B a; f (). For istace, whe f 3 ad [a] f 2 100, B a; f (5) 10 ad B a; f (7) 000 makes B a; f (5) B a; f (7) I order to prevet such cases, we remove the sequeces that coicide with a prefix of [a] f 2 from fb a; f ()g. Sice oe sequece is removed for each legth j if j<f, the obtaied biary umber B ~ a; f () is give by if M 2 (j; f) 0 L(j; f) <M 2(j +1;f) 0 L(j +1;f): (13) The legth ~ f () of B ~ a; f () is give by ~ f () j; if M 2 (j; f) 0 L(j; f) <M 2 (j +1;f) 0 L(j +1;f): (14) Some examples of B ~ a; f () are also show i Table I. ~B a; f () [ 0 M 2(j; f) +L(j; f)] j 2 ; if M 2 (j; f) 0 L(j; f) <M 2 (j; f) 0L(j; f) + N ~ 2 (j; f; a) [ 0 M 2(j; f 0 1) + L(j +1;f)] j 2 ; if M 2(j; f) 0 L(j; f) + N ~ 2(j; f; a) <M 2 (j +1;f) 0 L(j +1;f) (10) III. NEW RECURSIVE UNIVERSAL CODE OF POSITIVE INTEGERS Usig B ~ a; f () defied by (10), a ew recursive uiversal code of the positive itegers ca be defied similarly to (3) as follows: C a; f ( 0) B ~ a; f ( K ) B ~ a; f ( K01 ) 111 B ~ a; f ( 1 ) B ~ a; f ( 0 )[a] f 2 (15)

4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH where each k is give by k ~ f ( k01) 0 1 (16) TABLE II EXAMPLES OF C () ad K is the iteger k that satisfies k represeted recursively as follows: 1. C a; f () ca also be C a; f () ~ C a; f ()[a] f 2 (17) ~C a; f () ~B a; f (); if 1 ~C a; f ( ~ f () 0 1) B ~ a; f (); if >2. (18) We ote that sice B ~ a; f (1) is always equal to 0 or 1, the first segmet B ~ a; f (1) ca be omitted i the biary case. Some examples of C a; f () are show i Table II. Sice a prefix of ay B ~ a; f ( k ) does ot coicide with [a] f 2, [a] f 2 ca delimit the codeword. We ow derive upper bouds o the codeword legth l a; f () of code C a; f () defied by (15) (or (17) ad (18)). Let wf 3 () be K +1 or the iteger k that satisfies k 0for k recursively defied by (16). Note that wf 3 () is a mootoically icreasig fuctio of ad wf 3 () w3 (). The, the followig theorem holds. Theorem 1: l a; f () satisfies for ay that l a; f () log F 2(f )w 3 f () +c f + 2 () (19) ad l a; f () satisfies for ifiitely may that where l a; f () log (1 0 F2(f ))w3 f () +c f +2 2() (20) F 2 (f )0 ( f ) (21) c f 5(f 0 2) + + f +5F 2 (f) (22) 2() e 1+ (w3 ()01)( e) 01 4:7 : (23) Proof: We ote from (14) that i itegers with ~ () j, the smallest ad largest oes are give by M 2(j; f ) 0 L(j; f ) ad M 2(j +1;f) 0 L(j +1;f) 0 1, respectively. Hece, from (16), l a; f () 0 log 3 has a local maximum at the followig : K 1 (24) k01 M 2( k +1;f) 0 L( k +1;f) (25) 0 (26) while it has a local miimum at the followig : k01 M 2 ( k +2;f) 0 L( k +2;f) 0 1: (27) I the followig, we derive a upper boud of l a; f () for these two extreme cases. We first cosider the former case, i.e., the worst case. For k f 0 1, (25) becomes k f 0 (f 0 1): (28) Furthermore, for k f 0 2, k satisfies k (f 0 1) + (f 0 k 0 1) f 0 (f 0 1) 2 +1 ( f ) 0 (f 0 1): (29) Hece, for ay k 1; 2; 111,wehave k +1 ( k01 +(f 0 1)) + F 2(f ): (30) From (30), 1 is upper-bouded by 1 +1 ( +(f 0 1)) + F 2(f ) + 1+ f F 2 (f ) + e (f 0 1) + F 2(f ): (31) From (30) ad (31), 2 +1is bouded by 2 +1 ( 1 +(f 0 1)) + F 2(f ) + e (f 0 1) + F 2(f )+(f 0 2) + F 2(f ) log e (f 0 1) + F 2 (f )+(f 0 2) + F 2 (f ) 2 + e e (f 0 1) + F2(f )+(f 0 2) + F 2(f ): (32)

5 720 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 Repeatig such procedure, we obtai k +1 log k 2 + e log k ( e) 2 2 )(log k02 2 ) (F 2 (f )+(f 02)) + ( e) k01 2 )(log k02 2 ) 111( ) ( e) k 2 )(log k02 (f 01)+F2(f) 2 ) 111( ) log k 2 +(1+A k())f 2(f)+Ak+1(2 )(f 02)+Bk() where Ak() ad BK () are defied as (33) A 1 () 0 (34) holds for b e. Furthermore, A 01 () ad A () ca be bouded above as follows: A 01 () b log b 2 (log 02 2 )(log 03 2 ) b (log 02 2 )(log 03 2 ) b 2 + b 2exp 2 (2) + b 2 exp 2 (2) exp 2 2 (2) b exp 2 (2) 111exp 03 2 (2) b 1+ b b b b 2 1 i0 2b 4 0 b b 4 i (40) Ak() Bk() k02 j0 ( e) j+1 2 )(log k02 2 ) 1110j 2 ) ( e) k (35) 2 )(log k02 2 ) 111( ) : (36) The codeword legth i the local maximum (or worst) cases, say (), has the followig upper-boud from (16) ad (33) a; f A () 3 b + b2 2 + b 3 2 exp 2 (2) b 1+ b 2 + b 2 2 exp 2 (2) b 1+ 2b 4 0 b where iequality 3 holds because of b(4 + b) 4 0 b (41) a; f () 01 k0 01 k0 ~(k) +f (k+1 +1)+f ad for k 3. Hece, we have log 01 1 log 02 2 log 0k exp k02 2 (2) (k +1)+f (37) (log k 2 +(1+A k())f 2(f) + Ak+1(2 )(f 0 2) + Bk()) + f log w3 f ()F 2 (f) + Ak()F 2 (f ) + Ak+1(2 )(f 0 2) + Bk() +f: (38) We ote from [3, eq. (A-12)] that iequality O the other had, Ak() 4b 3(4 0 b) + 2b b(4 + b) b 4 0 b b 4 0 b Ak+1(2 ) 22 + b G2: (42) 3 +1 k2 w (2 ) Ak(2 ) Ak(2 ) (43) because w 3 (2 ) w 3 () +1ad A 1 (2 ) 0by the defiitio. Therefore, (43) is also bouded by 02 4b Ak() 3(4 0 b) (39) Ak+1(2 ) G 2 : (44)

6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH Furthermore, the sum of B k (), say 2(), ca be bouded by 3 2() B k () log e 2 (log 1 2 e) k01 1+ k2 2 )(log k02 2 ) 111( ) e 1+ (w3 () 0 1)( e) 01 (45) (46) 4:7 (47) where the last iequality holds because the secod term i the bracket of (46) has the maximum at 16. From (38), (42), ad (44), a; f () ca be bouded by a; f () log3 2 + w3 f ()F 2 (f ) + G 2F 2(f )+G 2(f 0 2) + + f + 2(): (48) Next we treat the local miimum (or best) case give by (27). I this case, we have for k f 0 2 ad for k f 0 3 k f 0 (f 0 1) 0 1 (49) k (f 0 1) + (f 0 k 0 2) f 0 (f 0 1) ( f ) 0 f: (50) This meas that, for ay k, k +2ca be bouded by k +2 ( k01 + f )+F 2 (f ): (51) Hece, i the same way as (33), we have k +2 log k 2 +(1+A k())f 2(f)+A k+1 (2 )(f 0 2)+2B k (): (52) Furthermore, from (37), (42), (44), ad (52), l (B) a; f () ca be bouded by l (B) a; f () ( k +1)+f ( k +2)0 w 3 f () +f (log k 2 +(1+A k())f 2 (f) + A k+1 (2 )(f 0 2) + 2B k ()) 0 wf 3 () +f log (1 0 F2(f ))w3 f () +G 2F 2(f ) + G 2 (f 0 2) + + f +2 2 (): (53) 3 A tight boud () 4 ca be obtaied by directly calculatig (45), where the secod term i the bracket has the maximum at 4. G 2 is bouded by G 2 4: < 5 because b e 1: Hece, we have from (48) ad (53) that a; f () log3 2 +F 2(f )w 3 f ()+5(f 0 2) + +f +5F 2 (f)+ 2 () l (B) a; f () log3 0 (1 0 F 2(f ))w 3 f () +5(f 0 2) + Fially, ote that ay satisfies (54) + f +5F 2 (f) +2 2 (): (55) k01 M 2( k +1;f) 0 L( k +1;f) (56) istead of (25). But (56) also iduces the same iequality (30) as (25). Hece (54) holds for ay. O the other had, (55) holds for ifiite may s that satisfy (27). Q.E.D. We ote from (21), (54), ad (55) that F 2(f ) ca be approximated as F 2 (f ) ( e)2 f, ad by settig f large, the coefficiet of w 3 f () i the worst case becomes very small while the oe i the best case becomes almost 01. Hece, we ca cojecture that l a; f () is shorter tha log 3 i large parts of the positive itegers. I the remaider of this sectio, we show that this cojecture is true by cosiderig a geeral case istead of the best ad worst cases. For a give k, k01 must be icluded i a regio R( k ) defied as R( k )f k01 : M 2( k +1;f) 0 L( k +1;f) k01 <M 2 ( k +2;f) 0 L( k +2;f)g: (57) We divide this regio R( k ) ito two regios, the worse regio R (W ) ( k ) ad the better regio R (B) ( k ), which are defied as R (W ) ( k )f k01 : M 2 ( k +1;f) 0 L( k +1;f) k01 < 1:5M 2( k +1;f) 0 L( k +1;f)g (58) R (B) ( k )f k01 :1:5M 2( k +1;f) 0 L( k +1;f) k01 <M 2 ( k +2;f) 0 L( k +2;f)g: (59) Sice the cardiality of R( k ), jr( k )j, is equal to M 2( k +1;f) for k f, the followig relatio holds: jr (W ) ( k )j jr( k )j jr(b) ( k )j jr( k )j Whe k01 2R (W ) ( k ), it satisfies 1 2 : (60) k01 M 2( k +1;f) 0 L( k +1;f): (61) Hece, i this case, we have from (30) that k +1 ( k01 +(f 0 1)) + F 2(f ): (62) Whe k01 2R (B) ( k ), it satisfies k01 1:5M 2( k +1;f) 0 L( k +1;f) (63) M 2 ( k :5; f) 0 L( k +1;f): (64) Therefore, we ca obtai similarly that k :5 ( k01 +(f 0 1)) + F 2(f ): (65)

7 722 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH 2000 For a give, lettig K (B) () be the set of k such that k 2R (B) ( k+1 ) ad lettig () be the ratio defied as () jk(b) ()j w 3 f () (66) the the codeword legth of satisfies from (37), (62), ad (65) that l a; f () ( k +1)+f ( k +1) k62k () + (( k +1+ 1:5) 0 1:5) + f k2k () log (() 1:5 0 F 2(f))wf 3 () + G 2F 2(f )+G 2(f 0 2) + + f + 2(): (67) Hece, whe is sufficietly large, the codeword legth l a; f () becomes shorter tha log 3 2 if () 1:5 0 F 2(f) > 0, i.e., () > 0 (1 0 20f ) 1:5 e 1:5 20f 2: f (68) where the approximatio holds for 2 0f 1. We ow show that (68) holds for almost all positive itegers. Assume that is uiformly distributed over the set of itegers satisfyig w 3 f () w for a give iteger w, ad a radom variable X k is defied as X k 0; if k 2R (W ) ( k+1 ) (69) 1; if k 2R (B) ( k+1 ) for such probability distributio. The, from the defiitio of R (W ) ( k+1 ) ad R (B) ( k+1 ),wehave where K w 0 1. Furthermore, sice PrfX K 0g 1 (70) jr (W ) ( k+1 )j jr (B) ( k+1 )j holds for ay k+1 f ad ay iteger icluded i R (B) ( k+1 ) is larger tha itegers i R (W ) ( k+1 ), we ca easily show that for w 3 ad 1 k K 0 1 PrfX k 0jX K 0;XK01 x k01 ; 111;X k+1 x k+1 g < 1 4 (71) PrfX 0 0jX K 0;X K01 x k01 ; 111;X 2 x 2 ;X 1 x 1 g 1 2 (72) hold for ay x K01 x K02 111x k+1 ad x K01 x K02 111x 2 x 1, respectively. 4 Obviously, PrfX k 1jX K 0;X K01 x k01 ; 111;X k+1 x k+1 g1 (73) 4 For simplicity, a rough upper boud 14 is used i (71) although PrfX 1jX 0;X x ; 111;X x g14 holds. also holds for ay k. Hece, a sequece x K01 x K02 111x 1 x 0 with m occurs with probability Pr (x) bouded above by Therefore, we have Pr(x) < Pr (N ) m w Pr K w02m 1 w010m 1 m w0102m 1 2m 4 m 1 m K k0 x k <w2 2 w : (74) k0 X k m < w m 20w (75) where N is the radom variable of. This meas from the law of the large umber for the biomial distributio that for ay fixed 0 < 12 lim w!1 Prf(N ) < 0g 0: (76) Sice it holds that 0 ( f ) 1:5 < 12 for f 3, we ca coclude that lim w!1 Prfl a; f (N) > log 3 2 N g lim w!1 Pr (N ) < 0 (1 0 20f ) 1:5 0: (77) Theorem 2: I case of f 3, the codeword legth of code C a; f () is shorter tha log 3 2 i almost all of sufficietly large positive itegers i the sese of (77). IV. CONCLUDING REMARKS We proposed a ew recursive uiversal code of the positive itegers, the codeword legth of which is shorter tha log 3 2 i almost all of sufficietly large positive itegers. Although we treated the biary case i the previous sectios, the results ca easily be exteded to r-ary code by usig log r, [] j r, [a] f r, M r (j; f ) rj 0 r (j0f) r 0 1 (78) ~N r (j; f; a) br j0f ac (79) istead of, [] j 2, [a] f 2, M 2(j; f), N ~ 2(j; f; a), respectively. For this case, we ca show, i the same way as the biary case, that a; f () ad l (B) a; f () are bouded as follows: a; f () log3 r + F r(f)wf 3 () +G r (f 0 2) + + f + G rf r(f) +r() (80) l (B) a; f () log3 r 0 (1 0 Fr(f))w3 f () +G r(f 0 2)+ + f + G r F r (f) +2 r () (81)

8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 2, MARCH where F r(f )0 log r 1 0 r 0f r 0 1 (82) same code C 0(K 0) to represet K 0 istead of 1 K 0 [9]. The the code have the followig structure: C 1 ( 0 )1 S 0[K S ] 0 2 [K S 01] [K 1] 0 2 [K 0] 0 2 [ K ] 0 2 [ K 01] [ 1 ] 0 2 [ 0] 0 2 : (86) G r r r log r e (r r 0 1)(r r 0 log r e) + rr log r e r(r r 0 log r e) + (rrr +(r r 0 r) log r e) log r e r(r r 0 log r e) log r e r r 0 log r e r () log r e (r r ) 2 r r rr +(r r 0 r) log r e r 1+ w3 () 0 1 log r (83) : (84) G r goes to log r e as r becomes large. We also ote that, as f becomes large, F r(f ) goes to logr (r 0 1), istead of zero, which is approximately equal to oe whe r is large. This correspods to the fact that i the r-ary case, the optimal code legth is give by logr 3 0 w3 () with < log r log r e < 0 [2], [3]. This meas that the codeword legth caot become shorter tha logr 3 for all large.5 Sice fuctio wf 3 () ad w3 () are mootoically icreasig fuctios of, the term of wf 3 () i (19) ad (20) becomes sigificat compared with costat terms for large. But, the rate of icrease is very slow. Hece, the overhead of delimiter legth f is more severe tha the effect of wf 3 () i moderate, ad the performace of the proposed code C a; f () is o better tha the kow codes i practical use. We ote that C a; f () ca be improved i the same way as Stout code [7]. But eve such modified code is ot suited for practical use. As we oted i Sectio II, Leveshtei W2 0 code [2] or Betley Yao code [5] have a code structure such that C 0 ( 0 )1 K 0[ K ] 0 2 [ K 01] [ 1] 0 2 [ 0] 0 2 (85) where 1 K is a sequece of 1 s with legth K 0 ad [ k ] 2 1[ k ] 0 2, i.e., [ k ] 0 2 is obtaied by deletig the MSB from [ k ] 2. Compared with C E ( 0 ) give by (3), the MSB s ad delimiter 0 are gathered as the prefix 1 K 0, which is the uary code deotig the recurrece umber K 0. Sice the uary code is iefficiet for large K 0, we ca use the 5 l () is a little shorter tha log i (81). But this is attaied by the loss of (80) which is larger tha log 0. Furthermore, S 0 i (86) ca be represeted by C 0 (S 0 ) istead of 1 S 0. By repeatig such recurrece arbitrarily fixed times t, code C t(0) ca be defied. But ote that C t ( 0 ) is a doubly recursive code while our code C a; f ( 0) is a simple recursive code. The evaluatio of the asymptotic performace for C t(0) is a ope problem. Fially, we ote that Leveshtei W 0 2 code ad Betley Yao code satisfy the lexicographic property. Recetly, Nakamura ad Murashima [10] showed that if their devised code C NM (K 0 ) is used istead of uary code 1 K 0 i (85), the legth of the code satisfies l() log w3 2() m + m + c (87) for ay give iteger m>0, ad the lexicographic property also holds i their code. REFERENCES [1] T. Amemiya ad H. Yamamoto, A ew class of the uiversal represetatio for the positive itegers, IEICE Tras. Fudametals, vol. E76-A, o. 3, pp , Mar [2] V. I. Leveshtei, O the redudacy ad delay of decodable codig of atural umbers, Probl. Cyber., vol. 20, pp , [3] R. Ahlswede, T. S. Ha, ad K. Kobayashi, Uiversal codig of itegers ad ubouded search trees, IEEE Tras. Iform. Theory, vol. 43, pp , Mar [4] P. Elias, Uiversal codeword sets ad represetatio of the itegers, IEEE Tras. Iform. Theory, vol. IT-21, pp , [5] J. L. Betley ad A. C. Yao, A almost optimal algorithm for ubouded searchig, Iform. Processig Lett., vol. 5, o. 3, pp , [6] S. Eve ad M. Rodeh, Ecoomical ecodig of comma betwee strigs, Commu. ACM, vol. 21, o. 4, pp , [7] Q. F. Stout, Improved prefix ecodigs of the atural umbers, IEEE Tras. Iform. Theory, vol. IT-26, pp , [8] T. C. Bell, J. G. Cleary, ad I. H. Witte, Text Compressio. Eglewood Cliffs, NJ: Pretice-Hall, [9] V. I. Leveshtei, private commuicatio, [10] H. Nakamura ad S. Murashima, Costructio method of positive iteger code by codig MSB strig of legth iformatio of existig codes (i Japaese), Tras. Iform. Processig Soc. of Japa, vol. 40, o. 4, pp , Apr