694 IEE

Algebraic Decoding Using Special Divisors

Iwan M. Duursma

Abstruct- The basic algorithm for decoding of algebraic geometric codes corrects up to ( d c - 1)/2 - g/2 errors where d c denotes the designed minimum distance of a code and g denotes the genus of a curve. The modified algorithm improves on this, but applies to a restricted class of codes. An extended modified algorithm that applies to all codes is formulated. It will correct up to ( d c - 1)/2 - s errors. s is called the Clifford defect of a curve. For curves with g 2 1, this defect satisfies 0 5 s 5 (g - 1)/2. The success of the algorithm depends highly on the curve that is used and the result is in the first place a theoretical result. To support the practical importance, two special cases are considered. All codes from hyperelliptic curves can be decoded up to the designed minimum distance. For plane curves that contain at least one rational point, it will be shown that s 5 (g + 1)/4.

Index Terms-Algebraic-geometric codes, decoding, special divisors.

I. INTRODUCTION In the basic algorithm (BA) [4], [7], an error-locator function

is determined. The error locations occur among the zeros of this function. Only when the set of zeros is not too large, the error vector can be determined. In the original set up two conditions guarantee the determination of an error-locator function with the number of zeros smaller than the designed distance. We weaken the two conditions and show that they yield correct decoding.

The modified algorithm (MA) [7] searches for error-locator func- tions of increasing degree and improves on the BA. The MA can be applied only to a restricted class of codes. The general case was left as an open problem [8, Remark 3.3.131. In this correspondence, we formulate the extended modified algorithm (EMA) that applies to all codes. The bound on error correction of the MA shows a defect that depends on the particular code. The bound on error correction of the EMA shows a defect that depends on the curve being used, rather than on the particular code.

We assume that the reader is familiar with the following no- tions: algebraic curves (rational function, rational differential, divisor, canonical divisor, genus), algebraic-geometric codes (residue code, functional code, designed parameters). All this is treated in [5] and summarized in [7].

11. O N THE BASIC ALGORITHM

For a description of the basic algorithm (BA), we refer to [4], [7], [6]. It can also be found in [8, ch. 3.31. We give a sketch of the BA and prove that two suitable conditions suffice for correct decoding.

Let C be a residue code Cn ( D , G). Say it has parity check matrix H . Let r = ( r p ) p E n denote a received word with error pattern e = ( e p ) r t n . Thus,

He' = Hrt. (1)

Manuscript received August 13, 1991; revised August 3, 1992. This work was supported by NWO, through Stichting Mathematisch Centrum. This work was presented in a talk at the 3rd International Workshop on Algebraic Geometry and Coding Theory, Marseille, France, June 17-21, 1991.

The author is with Eindhoven University of Technology, Department of Mathematics and Computing Science, P.O. Box 513, 5600 MB Eindhoven, The Netherlands.

IEEE Log Number 9204357.

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A n error-locatorfunction f is defined by the property

f ( P ) # 0 + e r = 0. (2)

The BA consists of finding a nonzero error-locator function f and then solving (1,2). The next lemma explains how f can be obtained. Let the divisor Q contain the error locations, that is, Q = xePfo P.

Lemma I : Let the divisor F with support disjoint from the support of D satisfy Cn (Q, G - F ) = 0. Let a bilinear form Sr be defined on the product space L ( F ) x L ( G - F ) as follows

sr ( f 5 9 ) = rp f (P )g (P) . (3) P E D

Then,

S r ( f . g ) = O . V ~ € E ( G - F ) S - ~ E L ( F - Q ) .

Proof: See [7], [6]. 0

To ensure that a nonzero error-locator function f can be determined the following conditions on the divisor F are sufficient

O(G - F - Q ) = 0, (4)

For the obtained f, the error e is a solution to (1,2). We show that (4,5) imply that e is in fact the only solution.

Lemma 2: Let f be an element of the left null space of a bilinear form Sr with conditions (4,5) satisfied. We know from Lemma 1 that the error locations & I , Q z , . . . , Qt occur among the zeros of f and we may write

with 0 5 E , E n ( D - Q * ) = 0. Thus, Q' contains all possible error locations. Solving for error values at these locations yields a unique solution.

Proof: In fact, let el and e2 be two possible solutions of (1,2). Then, for z = el - e2. z E C. In particular, z = ( w s r ( d ) ) p E D . for some d E CI(G - Q * ) . But,

E , Q 2 0 3 R ( G - Q * ) C G(G-Q' - E - Q ) .

and

( 6 ) . ( 4 ) 3 Q ( G - Q * - E - Q ) N n(G- F - Q ) = 0 .

Combination yields ij = 0, hence z = 0. 0

Remark 1: In [4], [6], [7], [8, Proposition 3.3.21 uniqueness is ensured by the restriction deg( F ) < d c . By the lemma this restriction is redundant. In particular the condition deg(G) 2 4g - 2 can be dismissed in [6] and the use of the definition of s ( H ) in the uniqueness proof can be avoided in [7], [8, Exercise 3.3.101,

Theorem 1 ([4, Plane Curves], [7, General]): Let C = Cn ( D , G ) be a residue code. The BA with F a divisor of degree [ ( d c - 1) /2 + g/2]with support disjoint from D will correct any error of weight up to L(dc - 1) /2 - g / 2 ] .

Proof: Conditions (4) and (5) are satisfied. 0

0018-9448/93$03.00 0 1993 IEEE

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111. SPECIAL DIVISORS

We present two well-known results and we define a parameter that will be used later to measure a defect in a bound on error correction. The parameter is investigated for hyperelliptic curves and for plane curves. For a curve k', let Zi be a representative of the canonical divisor class.

Theorem 2 (Riemann-Roch): For an arbitrary divisor E on the curve, we have

deg( li - E ) ___- d e g ( E ) ( I ( E ) - 1) = - (I(1i - E ) - 1). (7)

2 2

Proof: See e.g., [2, ch. 21, [3, ch. 41. 0

Definition 1: A divisor E is called special if it is effective and L ( 1 i - E ) # 0.

Theorem 3 (Cliflord): For a special divisor E on the curve, we have

Proof: See e.g., [ l , ch. 31, [3, ch. 41. 0

The curves considered in [3] are defined over an algebraically closed field. From this the result for finite fields follows. The proof in [ 11 for characteristic zero is similar for finite characteristic. Theorem 3 provides an upper bound for the dimension of a special divisor on a curve. To obtain a lower bound for the dimension of a special divisor we introduce the Clifford defect of a set. The defect measures the largest deviation from the upper bound.

Definition 2: For a curve A', let f be a finite set of divisors. We define the Clifford defect s ( f ) of the set f by s (0 ) = 0 and, for P # 0.

For our purpose, we consider sets E of special divisors that satisfy

f = { E ~ . E I : . . . E ~ ~ - : ! } . deg(Ec1 = i .

i = 0:...2g - 2. (10)

with g the genus of the curve (for g = 0 we set f = 8). One verifies that such a set exists, if and only if the curve has a rational point. For a fixed I, we write s = s(&) and we define subsets Eo. &I C & by

E E Eo <leg(€) E 0 (111od2). E E €1 CJ deg(E) E 1 (111otl2).

Also, let so = s(&o) and s1 = s ( & I ) . Lemma 3: For a divisor E , let s ( E ) denote s ( { E } ) =

d e g ( E ) / 2 - ( l ( E ) - 1) . A set f as in (10) can be modified without increase of its Clifford defect .s(P). such that it satisfies

I . s ( E , ) - s ( E , + , ) I = ~ / ~ . i = 0 . 1 : . . . 2 ~ - 3 . (11)

Proof: Besides ( l l ) , we may distinguish two cases:

1) s ( E , ) - s (E ,+ l ) > 1 /2 or l ( E L + l ) - ! ( E , ) > 1: 2) s (E ,+ , ) - s ( E , ) > 1 / 2 or l(Zi - E , ) - ! ( I < - E , + I ) > 1.

Let P be a rational point. As a modification in each case, we choose the following.

1) E, N € ,+I - P. E, 2 0. 2) E,+i = E, + P.

With each modification the nonnegative number s(Eo) + ~ ( € 1 ) + ' . . + s( E2g-2 ) will strictly decrease. Regardless of the order of the modifications, after finitely many steps condition (11) will hold. 0

Remark 2: For & as in (lo), the maximum in (9) is taken over nonnegative values by Theorem 3. With Theorem 2 we have

0 5 s 5 (g - 1) /2 . for g 2 1.

Also note that

so E 0 (mod 1) and .SI G 1 /2 (mod 1 ) .

A set E as in (10 , l l ) satisfies I s 0 - s I I = 1/2. We will later obtain bounds on error correction, t 5 (dc , - 1 ) / 2 - SO and t 5 ( d c - 1)/2-s l . respectively, for the cases of odd and even designed minimum distance respectively (Theorem 5).

Definition 3: We define the Clifford defect s ( S ) of the curve ,Y as

s(,Y) = miii{s(E) : E as in(lO)}.

In particular for elliptic curves we obtain s(,li') = 0. with I = {O}. From Remark 2, we conclude that decoding up to the designed minimum distance is also guaranteed for curves with ,$(,U) = 1 /2 (i.e., .so = 0 and SI = 1/2).

ProposLtzon 1: The curves with .s(.V) 5 1 / 2 can be classified as the curves of genus zero or one and the hyperelliptic curves.

Proof: Curves of genus g 2 2 with this property have a divisor E . E = EL E f. that satisfies

That is, the curve is hyperelliptic [3, p. 2981. Conversely, for a hyperelliptic curve ,Y the required set f = Eo U f, is defined by

with E as in (12) and P a rational point. 0

Curves with s ( ,Y) = 1 (i.e., so = 1 and s I = 1/2) allow decoding up to the designed minimum distance d c in case d c is even. Among these are the plane curves of degree 4.

Proposition 2: Let ,Y be a plane curve of degree 771 with genus g. With the assumption that the curve has a rational point and tti 2 4. a set I can be chosen with

In particular s ( . V ) 5 ( g + 1) /4 .

Proof: A construction for f is given in the Appendix. With g = ( r n - l ) [n i - 2 ) / 2 . we have ( m L - 4trr + 8 ) / 8 = ( g + 1 ) / 4 -

0 (111 - 3 ) / S and the inequality follows.

IV. MAIN LEMMA

Conditions (4), ( 5 ) are conflicting: the former is satisfied for a divisor F of sufficiently low degree, the latter is satisfied for F of sufficiently high degree. The following lemma allows one to choose divisors F of increasing degree, such that for at least one F in the sequence both conditions are satisfied.

~

696 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993

Lemma 4 (Induction Step): Let G, F , and Q be given divisors. Let E and F* be divisors satisfying

and

F* - G - F - E . (15)

Then,

L ( F - Q ) = 0 3 R ( G - F' - Q ) = 0.

Proof: We assume R ( G - F* - Q ) # 0. say it contains w # 0. and will deduce L ( F - Q ) # 0. So let

(U) = G - F* - Q + E', E* 2 0 .

Using the definition of F* (15), we obtain

( d ) - F + E - Q + E', F - Q - I i - E - E * , (16)

where I< represents the canonical divisor class. Since E' 2 0, it suffices for L ( F - Q) # 0, to prove

deg(E*) < l ( K - E). (17)

Substitution of (16) in (14) yields

(g - 1) - deg(1i - E - E * ) < Z(E). deg(E*) < Z(E) - d e g ( E ) + g - 1,

deg (E*) < Z(E) - deg(E) /2 + deg(1i - E ) / 2 . d e g ( E * ) < Z(K - E ) ,

where we apply Theorem 2 (Riemann-Roch). Thus, we have

For a residue code C, we make the bound on error correction that is contained in (14) explicit.

Lemma 5 (a Bound on Error Correction): Let C = CO ( D . G ) be a residue code. It has designed minimum distance d c = deg(G - I<) . Also, let t = deg(Q). Then, (14) in Lemma 4 is equivalent to

proven (17). U

d c - 1 d e g ( E ) d e g ( F * - F) - 1 2 + ( l ( E ) - 1) - - - t < - 2

Proof: Note that (15) implies that the right hand side of (18) is a natural number. We will obtain (14) from (18). To this end, we first multiply and then substitute for the parameters t and d c ,

2 t 5 ( d c - 1) + 2(Z(E) - 1) - d e g ( E ) - deg(F* - F ) + 1 . 2 deg(Q) < deg(G - l i ) + 2 ( l ( E ) - 1 ) - d e g ( E ) - d e g ( F * - F ) .

Next, we use (15),

2 deg(Q) 5 2(1(E) - 1) + d e g ( G - I< - E ) - deg(G - 2 F - E ) . deg(1i) - 2 d e g ( F - Q) I 2(1(E) - 1).

(g - 1) - d e d F - Q) 5 ( l ( E ) - 1).

This clearly yields (14). We have equivalence at all steps. 0

v. ON THE MODIFIED ALGORITHM

Theorem 4 (Modified Algorithm [7]): Let C = C Q ( D , G ) be a residue code, with G = a H . H an effective divisor, and h = deg(H) . Let

- s ( H ) = max

Then, a received word with error pattern of weight t , t 5 ( d c - 1)/2 - s ( H ) , can be corrected by successive applications of the BA with F = H , 2 H , . . . . In particular, F = bH of lowest degree such that ( 5 ) holds will satisfy (4).

Proof: A slightly improved bound is given in Proposition 4. U

Theorem 5 (Extended Modified Algorithm): Let C = Cn ( D , G ) be a residue code, defined with a curve X , with odd designed minimum distance d c = 2e + 1. Let & be a set of special divisors on X as in Definition 2 and let Eo be the subset of divisors of even degree, say EO = { E o . E l : . . , E , - i } and S O = s(&o):

Also, let F = { Fo, Ft .. . . , Fg} be a set of divisors on A', with deg(F0) = e. Fo n D = 0 and

Then, a received word with error pattern of weight t , t 5 ( d c - 1)/2 - so, can be corrected by successive applications of the BA with F = Fo . Fl . . . . , Fg . In particular, F = F, of lowest' degree such that (5 ) holds will satisfy (4).

Proof: With (19) we have, for i = 2.3:.. ,g .

deg(F, + E t - l ) = deg(G - Ft-i) = deg(F,-z + EI-z) .

deg(F, - E - 2 ) = dt=g(E,-s - E,- i ) = 2.

With deg( Fo ) = e and deg( F, ) = P + 1 we obtain

d e g ( F , ) = e + i , i = O , l , . . . , g .

Let Q denote the divisor of all the error locations. For FO we have, with deg(Q) = t < e - so and SO 2 0.

deg(G - FO - Q) 2 deg( l<)+2e+l -e- (e - S O ) 2 d e g ( K ) + l .

Thus, R ( G - FO - Q ) = 0. Next, we prove that, for i = 0, l:...g - 1.

We now use Lemma 4. All conditions but (14) are trivially fulfilled. We have

We may apply Lemma 5. This proves the induction (20). For the termination of the algorithm, we have for Fg

0 Hence L(Fg - Q) # 0.

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Remark 3 (Even Designed Minimum Distance): 1st C' = C'rj( D . G) have even designed minimum distance I / ( , = 2f ' + 2. The following modifications apply to Theorem 5 . We consider the subset EI of E of divisors of odd degree, say El = { E l . € 2 . . . . E,, I } and SI = s(E1 ) :

t l t y 5 € , ) = 2,q - 1 - 2;.

t l t g ( E , ) / 2 - ( [ ( E , ) - 1 ) 5 S I . i = 1 . 2 :...!/ - 1

Also, let F = {FI. F 2 . . ' . . F , } . with deg( F t ) = t + 1. Fl n D = fl and (19) for i = 2.3.. . . . q. Then, a received word with error pattern of weight t . f 5 ( d r , - 1 ) / 2 - S I . can be corrected by successive applications of the BA with F = FI . FL. . . . . F,,. The proof follows Theorem 5.

Proposition 3: For the following curves, the EMA is t-error- correcting. Rational, elliptic and hyperelliptic curves:

t 5 [ ( ( I ( . - 1) /2J .

Plane curves of degree t t i :

f 5 L(d(. - 1 ) / 2 - ( t t l - 1 ) i t t J ~ :1)/8].

Proof: For the first bound we use the set F as in Proposition I . 0

Proposition 4; We claim that Theorem 4 holds with the following

For the second bound we refer to the Appendix.

H may be any divisor and s ( W ) can be defined as

Proof: The improvement is obtained through a different unique- ness proof (Remark 1). Note that the bound on error correction, t 5 ( d r . - 1 ) / 2 - . s ( H ) . always represents an integer. We use the line of proof in Theorem 5 . Clearly, we may add F = 0 at the beginning of the sequence. The following are trivial: <?(G - 0 - ( J ) = 0 and L ( u H - Q ) # 0. For the analogue to the induction (20) , we will apply Lemma 4 with F' = F + H . Then, E = C; - F - F* runs through

E = (0 - 1 ) H . ( ( I ~ 3 ) H . ' ' ' .

And, for E = i H .

t lrg(E)/2 - ( [ ( E ) - 1) = ( ; I / ) / ? - ( l ( i H ) - 1)

5 .\(If) - ( h - l ) /? .

Thus we have, at all steps,

tlcy,(F* - F ) = h .

and

tlc,g(E)/2 - ( [ ( E ) - 1) 5 s ( H ) - ( I / - 1 ) / 2 .

and

We may apply Lemma 5 . This proves the induction. U

APPENDIX

A. Proof of Proposition 2

Proof We consider the plane curve .I' of degree t t / and genus $1 = (nr - l ) ( m - 2 ) / 2 . With the assumption J J J 2 4. we will construct a set t' with

.(F) = ( u i 2 - A J J J + 8)/8. if J J I E (1 ( i i i o (12 ) .

{ ( J J J ' - At t t + 7)/8. I f tJ / E 1 ( I l l O ( 1 2 ) . (21)

Thus, let L be an intersection divisor of the curve with a line. Also, let B( be an effective divisor of degree d e g ( B ? ) = i . for I = 0.1. ' ' . t t ) - I . We define the set :* as the set of divisors E , that satisfy one of the following:

E = I I L + l3,. 0 -< t i < ~ :I m t l 0 5 i 5 t t i - ( U + 2 ) .

E = j r i + 1 ) L - U , , , - , . 0 5 r i < J J f - :3 it i l( l

t t t - ( t / + '11 < I < t t r . E = i t t l - 3 ) L .

One verifies that :- contains divisor., E , with degree d e g ( E ) in the range: 1) 5 t l ( ,g (E) 5 ( t u - 3)fJt = ?!I - 2. To each degree in the range there corresponds a unique divisor. With J J / 2 4 we note: in the calculation of the maximum .si:.), w e may consider E # ( t t r - 3 ) L . We have / i r i L ) = ( ( I + 2 ) ( ( / + l ) /? . and thus for [ ( E ) .

/ ( t J f . + B , ) 2 (0 + ? ) ( < I + I)/?. and

/ ( ( t i + l ) L - B , , , - , ) 2 i , / + ~ ) ( ( / + 2 ) / ~ - ( t t / - i ) .

F o r d e g ( E ) / ? - ( ( ( E ) - l ) w e o h t a i n . w i t h i = t t c - ( n + 2 ) F b . b 2 0.

(leg( E ) --ill + ( 1 1 1 - A ) ( / + I I J - 2 - (1

> ' (22) ~- ,, ( [ ( E ) - 1 ) 5

Thc maximum of the right hand side is obtained for ( I = ( t t i - A ) / ? . / J = I ) . Looking for integer solutions, we see that the maxima are obtained for

t t r - 4 j/2. 0). t i / - 3) /2 . l ) ) .

if I J ~ E 0 ( i i i o t l 2) . if t t t E 1 (111otl2).

( t / . h ) = {i: Substitution in (22) yields (21). 0

B. Proof of Proposition 3

Proof? From (21) we obtain the inequality

( J J t ~ I ) ( 111 - 3 ) / b 5 h - I/?.

The gap is at most 1/8. Also s - 1 / 2 5 .so and s - 1/2 5 s 1 .

where one inequality is tight and the other shows a gap of 1 /2 . For the cases of odd and even designed minimum distance the following represent tight integcr bounds for decoding with the EMA (Theorem 5 and Remark 3)

t 5 ( r l c - 1 ) / 2 ~ s,,. for ( I ( o t l t l .

t 5 [ , lc . - 1 ) / ? - S I . f(1r ( I ( . ove11

Hence, one verifies by combination of the inequalities that in both cases

f < - ( ( l r ' ~ I ) / ? - ( I t 1 - 1 ) ( I l l - 3) /6 .

where the gap is at most .>/b. 0

E. Arbarcllo. M . Cornalha. P . A . Griffith4 and J . Harris, G e t i m q of Algehi-uit, Crrrwc 1.

Chevalley. /ri/rodirc./ion / ( I /he Tlicorv (~/"A/,++ruit. Funcrion.5 of Oire riuhlc. New York: AMS, I 95 I .

R. Hartshornc, Algrhruic Gcwrnc, / r~ . NCH Yoi k: Springer-Verlag. 1977. J . Justesen. K. J . Larwn. i l . E. Jcnsen. A. Havcmose, and T. Hoholdt, "Construction and decoding of a class of algebraic geometric codes." I E t t Trufis. / i i f imn. Theor-v, vol. 35. pp. X I 1-821, July 1989. J . H. Van Lint and (;. van der ( ieer. / i i /roducf i~~i i /o Coding Theor\ uiid A/,yrhruic Geonic~rrj. R. Pellikaan. "On a decoding algorithm tor codes on maximal curves." / k . tL Truirc. Irifiwm. Thhcwry. vol. 35. pp. 1228- 1232, Nov. 198Y.

New York: Springer-Verlag, 1985.

Hasel: Birkhiuser Verlag, 1988.

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P I

A.N. Skorobogatov and S.G. Vlsduf, “On the decoding of al- gebraic-geometric codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 1051-1060, Sept. 1990. M. A. Tsfasman and S. G. Vlsdut, Algebraic Geometric Codes. Dor- drecht: Kluwer. 1991.

Exact Analysis of the Lempel-Ziv Algorithm for I.I.D. Sources

Tsutomu Kawabata

Abstract-A new analysis shows that, when we apply the Lempel-Ziv incremental parsing algorithm to an i.i.d., source with probabilities p , , I = 1.. . . . 111, the expected length ElWt 1 of the tth parsed segment Wt is given by ( A )E!,=, ( - 1 ) ’ I - l nYz2 (1 - pf ) , with the null product being 1. It is also shown that limt-, ElWtl/ log t = H ( S ) - ’ for the same source with the entropy H ( S ) . This gives a variable-to-fixed length (VF) scheme version of the Ziv-Lempel universal coding theorem.

Index Terms- Lempel-Ziv algorithm, first passage time, functional equation, probability generating function, variable-to-fixed length (VF) source coding.

I. INTRODUCTION

The Lempel-Ziv incremental parsing (LZIP) for a sequence of discrete alphabet is to parse the sequence such that each newly parsed segment is the longest one previously parsed plus one new symbol. This enables us to represent the newly parsed segment from the parsing history in a compact, reproducible and computationally efficient way. This fundamental idea appeared in one ([l]) of the series of celebrated papers, and it has been used in a variety of universal data compression systems (e.g., [2], [3] ) .

In this correspondence, we analyze the performance of LZIP for independent identically distributed (i.i.d.) source by a new method. Its merit is that we can represent the expectation of the length of the parsed segments exactly in a closed form. From this formula we can calculate the data compression ratio of LZIP in a framework of the variable-to-fixed length (VF) coding scheme.

The performance of LZIP is analyzed in Ziv and Lempel’s original paper [ l ] and in the following researches ([l], [6], [SI) as a fixed-to- variable length (FV) coding scheme. Their assumption for the source is far weaker than i.i.d. since it includes the stationary and ergodic sources. Moreover, it is applicable to an individual sequence. One shortcoming is the reliance on the resolvability of Ziv’s inequality (cited in [4]), which relates the probability of a sequence to the number of its distinct parsed segments. It would be more natural to seek an exact and compact formula, which is done in this correspondence. Moreover, since our scheme is VF the definition of the rate is different. Relations between the two definitions should be clarified. Starkov and Tjalkens [9] independently attack the same problem but with a different approach, although they fail to obtain a closed form solution.

Manuscript received February 26, 1992; revised May 11, 1992. The author is with the Department of Communications and Systems,

IEEE Log Number 9204099. University of Electro-Communications, 1-5-1, Chofushi, Tokyo 182, Japan.

In Section 11, we repeat the definition of LZIP for notational clearance and state the two main results. The first, Theorem 2, gives the closed form solution previously mentioned. We give an example of Theorem 2 at early stages of parsing. The formula is simple, but its evaluation is not so direct since explosively growing alternating terms appear in the sum. The second result, Theorem 3, compensates this and gives the first-order asymptotics of the expected length of the tth parsed segment. It also ensures the asymptotic optimality of LZIP. We prove Theorem 2 in Section I11 and Theorem 3 in Section IV.

11. PRELIMINARIES AND THE STATEMENT OF MAIN RESULTS

LZIP is described as follows. Let A = {a1 , . . . , a,} be the source alphabet and z E Am an infinite input string. Let initially n = -1 and T-1 = { A } , where X represents the empty string. We iterate the following: After incrementing n by unity, let wn be the unique maximal prefix of z in T,-1 and then delete wn from z’s head. Delete w,, from T,-, and add w,a to the same set for all a E A renaming the resultant set as T,.

After n becomes positive, each repetition is called a parsing. A sequence of parsings produces a sequence of parsed segments w 1 , w 2 , . . . together with the sequence of parsing sets, TI, Tz, . . .. These are also called parsing trees since we can regard each of them as a complete set of strings, i.e., the leaves of a complete m-ary tree. This justifies the notation TI for the set of all strings defined by the inner nodes of this tree. The ordinal of wn in Tn-l is a compact but sufficient information to reproduce w n , provided we know of T,-l. Obviously, the ordinal enables the decoder to maintain the parsing tree for the next reproduction. To apply the algorithm for FV coding scheme, we need a modification to handle the termination. Throughout the paper the base of logarithm is two. A string is denoted either with a bold face like z or by indicating a lower and upper index like in zk for the subsequence X k , . . . , 21 .

Theorem I (Ziv-Lempel [l], an improved Proof by @ner and Ziv 161 (in [4])): Assume that the source is ergodic and stationary with entropy H ( X ) . By an appropriate use of a comma-free code for ordinals, and by a modification for the termination, the LZIP encodes the data sequence z; into a binary code of length L(z; ) , which satisfies

l imsup - L(-y‘) = H ( X ) (with probability 1). (1)

In [4], linlsup L ( X ; ) / n was shown to be an upper bound for H ( X ) with probability one. The equality can be shown if we note that linin+m E[L(x ; ) / n ] = H ( X ) , which is obtained by the use of Shannon’s bound for the average length of an a-block code and by the definition of the entropy.

In this correspondence, we are concerned with a more detailed performance of this algorithm. The code produced by the LZIP is essentially a VF code since the number of previous parsings uniquely determines the set of the indices.

We define the rate of a general VF code. Let C C U E 1 A k be a complete prefix set defining a VF code. One time performances of a VF may not be compared in general to those of a FV code. However, consider using C repetitively. Then any source sequence z E Am can be decomposed uniquely as z = S:~+~Z::+~ ... such that z::-,+~ E

C , where 0 = TO < TI < .... If N ( n ) = max{i : 7% 5 a}, then s; is determined by at most x(72) + 1 codewords. Hence, the information in bits per symbol required to represent z’; will be at least N ( n ) log IC(/n and at most ( N ( n ) + 1) log ICl/n. If we assume the

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001&9448/93$03.00 0 1993 IEEE