RECHERCHES Womcodes constructed with projective geometries «Womcodes» construits à partir de géométries projectives Frans MERKX (') École Nationale Su

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1 Wmcdes cnstructed with prjective gemetries «Wmcdes» cnstruits à partir de gémétries prjectives Frans MERKX (') Écle Natinale Supérieure de Télécmmunicatins (ENST), 46, rue Barrault, PARIS Étudiant à l'université de Technlgie à Eindhven (Pays-Bas) dans le dmaine des mathématiques discrètes. SUMMARY We cnsider strage media which cnsist f a number f write-nce bit psitins (wits). A wit initially cntains a "0", that may be irreversibly verwritten with a "1". It was shwn by Rivest and Shamir [5] that, by cding techniques ne can reuse such a write-nce memry (wm) up t a very high rate. We present tw new cyclic wmcdes, based n PG (2,2) and PG (3,2) respectively, which attain the RS-bund. These cdes can be decded with a decding algrithm fr Hamming cdes. Sme ther high-rate wmcdes, derived frm thse abve, are discussed. Errr-crrecting cdes, finite gemetry, numerical strage media. KEY WORDS RÉSUMÉ Nus cnsidérns des mémires cnstituées de psitins permettant l'écriture irréversible d'un bit (wits). Un wit cntient initialement un zér, qui peut être définitivement transfrmé en un. Nus utilisns des techniques de cdage pur réutiliser ces mémires à écriture unique avec un rendement élevé. Cdes crrecteurs d'erreurs, gémétrie finie, mémires numériques. MOTS CLÉS (') During this research, the authr was with the Écle Natinale Supérieure de télécmmunicatin, Paris. 1227

2 WOMCODES CONSTRUCTED WITH PROJECTIVE GEOMETRIES CONTENTS 1. Intrductin 2. Ntatin 3. A < 7 > 4 /7 wmcde 4. Mre wmcdes frm prjective gemetries 5. Extensins Cnclusin Acknwledgements References 1. Intrductin We cnsider strage media which cnsist f a number f write-nce bit psitins (wits). A wit initially cntains a "0", that may be irreversibly verwritten with a "1". We call such a strage medium a "writence memry" r wm. Examples f wms are punched cards r digital ptical disks. In their pineering paper n this subject, Rivest and Shamir [5] shwed that it is pssible t use a wm several times, by using "wmcdes". They gave many examples f wmcdes, and shw that the "capacity" (defined later) f a wmcde is greater than the number f wits. They als derive asympttic results fr this capacity. The fllwing cding scheme was their prime "mtivating example". Example 1. 1 : We write tw times 2 bits in a memry f 3 wits, as fllws : Message x First writing r (x) Secnd writing r' (x) This scheme must be interpreted as fllws : The first time we receive a message x, we write r (x). Later, we will receive fr the secnd time a message, say y. If x =y we dn't change the memry, if x*y we change the memry state t r' (y), by nly changing 0's t 1's. Withut cding, we wuld have t use 4 wits. Remark that after the secnd writing we lst the infrmatin n the first message. 2. Ntatin We call a cding scheme that uses n wits t represent t times ne ut f v messages (i. e. write nce and change t - 1 times) a < v > t/n wmcde. Mre generai, with a < v l, v 2i..., vt >/n wmcde the message in the i'th generatin can lie ne f a set V, f vi messages. Such a wmcde must have the fllwing prperties : 1. Each memry cntent that ccurs must determine uniquely the last received message, and 2. fr each memry cntent x e { 0, 11" ccuring in the s'th generatin (s < t) it must be pssible t encde all sequences f messages K+1,.., m t ), mi e V i, such that x, :g x, 1 <... < x t (cmpnentwise). The wmcde is determined by giving fr all pssible memry states and new messages the new memry state (update functin [5]). w (< v t, v 2,..., vt >) dentes the least n fr which a ( v l, v 2,..., vt i/n wmcde exists, and, f curse, w(<v> t)=w (<v, v,..., v>). Rivest and Shamir [5] derive a lwer bund fr w (< v > 1 ). This can be easily extended t a lwer bund fr w (< v 1, v2,..., v t i), see [4], which we shall refer t as the RS-bund. The capacity C and the rate R f a < u t, v 2,..., vt >/n wmcde are defined as C :=1g (v 1.v 2 v t), R : =C/n. Fr example, the < 4 > 2/3 wmcde f example 1. 1 has C=4 and R= (= bit per wit). The fllwing tw examples f wmcdes are described in [5]. Example 2. 1 : A < 5 > 3 /5 cyclic wmcde (Rate= ), cnstructed by D. Klarner. Message 1 is represented by in the first generatin, either r in the secnd, and ne f 01111, r in the third generatin. Message i, 1 <= i <_ 5 is represented by a cyclic shift ver i-1 psitins f the wrds fr the first message (since 5 is prime they are all distinct). Example 2.2 : A < 7 >'/7 cyclic wmcde (rate = ), cnstructed by D. Leavitt, by extending the methd f example 2. 1 (t appear). We will present a new cyclic < 7 >'/7 wmcde. The first, secnd and furth generatins are equivalent t these f Leavitt's cde, but the thid generatin is different (private cmmunicatin). We will use a prjective gemetry PG (2, 2) r 2-(7, 3, 1) symmetric blck design (Steiner triple system) r Fan plane. 1228

3 L These are different names fr the same bject, described in Figure Fig The Fan plane. This picture shuld be interpreted as fllws : The plane cnsists f seven pints (numbered 1, 2,..., 7) and seven lins, each cntaining three pints (the six lines tgether with the circle in Figure 2. 3). Remark that any pair f pints is n exactly ne fine, and any tw lines intersect in exactly ne pint. With the enumeratin as in Figure 2. 3, the incidence vectrs f lines with the pints f the plane are the cyclic shifts f In what fllws, we shall identify the seven pints f the plane with the wits f a seven-wit memry. Figure 3. 1 shuld be interpreted as fllws : pssible memry states f the wmcde are described by their crrespnding cnfiguratins [The encircled pints in (3. 1)]. Since there are seven messages, they are identified with the seven pints in the Fan plane. Fr each cnfiguratin the message (pint) represented is indicated by an arrw. Using the prperties f the Fan plane given after (2. 3) it is easy t check that Figure 3. 1 describes indeed a < 7 >'/7 wmcde : 1. Each memry state determines uniquely a message pint, and 2. Fr each cnfiguratin in generatin i, and fr each message pint received, it is pssible, by adding ne r tw pints, t find a cnfiguratin in generatin i+1 representing the received message pint. (NB : if the same message is received twice, the memry state is nt changed.) Example 3. 2: Suppse we receive the message sequence 2, 5, 3, 7. Using the wmcde f (3. 1), we btain the fllwing sequence f cnfiguratins representing them: As a cnsequence, every memry cntent can be identified with a cnfiguratin f pints in the plane (i. e. the pints fr which the crrespnding wit cntains a 1). 3. A < 7 >'/7 wmcde The < 7 >'/7 wmcde which we prpse is described in terms f cnfiguratins f pints in the Fan plane in Figure 3. 1 belw. generatin Ouater cnfiguratins Crrespnding t the sequence f memry cntents , , , O (a pint) DECODING 3 4 (tw pints) (a line and a pint) C. r c (a plane with ne r tw pints missing) Fig A descriptin f a < 7) /7 wmcde. Fr the decding, we use the fact that the cdewrds f the [7, 4] binary perfect Hamming cde are all the linear cmbinatins mdul 2 f the lins f the Fan plane (see, fr instance [3]). S the cde wrds are 0', lines, symmetric differences f tw lins, and the whle plane. Cnfiguratins f the wmcde in (3. 1) are never Hamming cdewrd, s they are at Hamming distance 1 frm exactly ne Hamming cde wrd. Mrever, by inspectin f (3. 1) we sec that this Hamming cdewrd is btained by adding (md 2) the message pint t the cnfiguratin. S, if we dente the memry cntent as a Hamming cdewrd, the errr vectr yields exactly the message 229

4 WOMCODES CONSTRUCTED WITH PROJECTIVE GEOMETRIES (i. e. using syndrme decding, each message crrespnds with ne f the seven pssible nn-zer syndrmes). Fr sme f these cnstraints the exact value f t is determined. If turns ut that, using nly the minimal weight cnstraint, des nt always (and prbably nly fr n = 3 r n = 7) yield the maximal t. 4. Mre wmcdes frm prjective gemetries Example 4. 1 : Cnsider the prjective gemetry PG (3, 2). It cntains 15 pints, 35 lines f 3 pints and 15 Fan planes f 7 pints. The [15, 11] Hamming cde can be seen as the cllectin f all linear cmbinatins md 2 f lins in PG (3, 2). Nw the apprach f sectins 3 can be generalised : it is pssible t cnstruct a < 15 >'/15 wmcde which can be "decded" by the methd described in sectin 3, i. e. the message represented by each memry state in the wmcde is just ils Hamming cde errr, and is btained by cmputing the syndrme (see [4] fr details). Example 4. 2 : Fix a line in the Fan plane. The cnfiguratins f the first tw generatins in (3. 1), restricted t thse n this line, describe a < 3 > 2 /3 wmcde. Since a line is a PG (1, 2), this wmcde culd be cnsidered as the first cde f a class f wmcdes, based n PG (n, 2), all having the prperty that they can be dented with syndrme decding fr the Hamming cde. Of curse, the secnd cde f this class is the cde in sectin 3, the third is (4. 1). These three cdes are ptimal in the sense that w(<3>2)>_3, w(<3>3)>_4, w(< 7 > 4)>7, w(<7>5)>_8, w(<15> 7 )>_15, w(<15>')>_16, all by the RS-bund. The next cde in this class is a < 31 >731 wmcde. The RS-bund yields w (< 31 > 14) >_ 31, w (< 31 > 15) >_ 32. We have cnstructed a < 31 > 10 /31 wmcde, s that sme sequences f length 11 cannt be encded. Hwever, we think that by further selectin f the cnfiguratins it is pssible t cnstruct at least a < 31 > 12 /31 wmcde. An imprtant feature f the cdes described is that messages crrespnd t a cset f a linear errrcrrecting cde with (n-k) x n parity check matrix H. Encding a new message m in a memry with state x c { 0, 1 }n is equivalent t finding a y e { 0, 1 }" such that m =y. HT and y >_ x, cmpnentwise. This is als described in [2], tgether with a dynamic prgramming algrithm fr finding a y with minimal weight. Hwever, ther cnstraints fr y than having minimal weight, can be psed, such as "y must be ne f the cnfiguratins f (3. 1)". In [4] sme cnstraints are described t cnstruct < 2n -k _ l > t/2 n-k - l wmcdes based n the crrespnding Hamming cdes. 5. Extensins Fr mre details f these extensins, see [4]. Extensin 5. 1 : Cnsider again the cde f (4.2). If we allw a 4-th message, represented by the empty set and the whle line, in the first resp. secnd generatin, a < 4 > 2/3 wmcde is btained. This is the wmcde f example Extensin 5.2: Cnsider the cde f sectin 3. If, in the first, third and furth generatin, we allw an cighth message, represented by the empty set, a line r the whle plane respectively, we btain a < 8, 7, 8, 8 )/7 wmcde (rate is ). Extensin 5. 3 : Adding t the cnfiguratins in the third generatin all thse cnsisting f three nnclinear pints, it is pssible t represent three mre messages, giving an < 8, 7, 11, 8 >17 wmcde (rate is ). 11 is best pssible here. Extensin 5. 4 : By extending the memry f the cde f sectin 3, respectively example 4. 1 with ne wit, it is pssible t cnstruct an < 8, 14, 11, 8 >/8 (rate is ) and a < 16 >'/16 (rate is ) wmcde, respectively. Cnclusin We have cnsidered memries (wms) which cnsist f a number f Write-Once bit psitins (wits). Using prjective gemetries, we have cnstructed cdes, which make it pssible t use these wms several times. It turned ut that the message, represented by the memry-cntent, can be seen as the syndrme f the binary Hamming cde. Acknwledgements We wuld like t thank Henk Hllmann fr his great help

5 REFERENCES [1] A. FIAT and A. SHAMIR, Generalized Write-Once Memries, IEEE Transactins n Infrmatin Thery, IT-30, 1984, n 3, May 1984, pp [2] C. HEEGARD, An efficient encder fr ptical disk cdes, preprint. [3] J. H. VAN Lira, Intrductin t cding thery, Springer Verlag, New Yrk, [4] F. MERKX, Cdes fr Write-Once Memries, Rapprt Interne, Écle Natinale Supérieure des Télécmmunicatins, Paris, [5] R. L. RIVEST and A. SHAMIR, (1982), Hw t reuse a Write-Once Memry, Infrmatin and Cntrl, 55, 1982, pp