ieee transections on information theory - technion

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 3, MARCH 2013 1573

Tilings With -Dimensional Chairs and TheirApplications to Asymmetric Codes

Sarit Buzaglo and Tuvi Etzion, Fellow, IEEE

Abstract—An -dimensional chair consists of an -dimensionalbox fromwhich a smaller -dimensional box is removed. A tiling ofan -dimensional chair has two nice applications in somememoriesusing asymmetric codes. The first one is in the design of codes thatcorrect asymmetric errors with limited magnitude. The second oneis in the design of cells -ary write-once memory codes. We showan equivalence between the design of a tiling with an integer lat-tice and the design of a tiling from a generalization of splitting (orof Sidon sequences). A tiling of an -dimensional chair can definea perfect code for correcting asymmetric errors with limited mag-nitude. We present constructions for such tilings and prove caseswhere perfect codes for these type of errors do not exist.

Index Terms—Asymmetric limited-magnitude errors, lattice,-dimensional chair, perfect codes, splitting, tiling, write-oncememory (WOM) codes.

I. INTRODUCTION

S TORAGE media that are constrained to change of valuesin any location of information only in one direction were

constructed throughout the last fifty years. From the older punchcards to later optical disks and modern storage such as flashmemories, there was a need to design coding that enables thevalues of information to be increased but not to be decreased.These kind of storagemedias are asymmetric memories.Wewillcall the codes used in these medias, asymmetric codes. Some ofthese memories behave as write-once memories (or WOMs inshort) and coding for them was first considered in the seminalwork of Rivest and Shamir [19]. This work initiated a sequenceof papers on this topic, e.g., [6], [9], [10], [32], and [37].The emerging new storage media of flash memory raised

many new interesting problems. Flash memory is a nonvolatilereliable memory with high storage density. Its relativelylow cost makes it the ideal memory to replace the magneticrecording technology in storage media. A multilevel flash cellis electronically programmed into threshold levels that canbe viewed as elements of the set . Raisingthe charge level of a cell is an easy operation, but reducing thecharge level of a single cell requires to erase the whole block

Manuscript received April 18, 2012; revised August 12, 2012; accepted Oc-tober 15, 2012. Date of current version November 12, 2012; date of currentversion February 12, 2013. This work was supported in part by the Israel Sci-ence Foundation, Jerusalem, Israel, under Grant 230/08.The authors are with the Department of Computer Science, Technion—Israel

Institute of Technology, Haifa 32000, Israel (e-mail: [email protected];[email protected]).Communicated by O. Milenkovic, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2012.2226925

to which the cell belongs. This makes the reducing of a chargelevel to be a complicated, slow, and unwanted operation. Hence,the cells of the flash memory act as an asymmetric memory aslong as blocks are not erased. This has motivated new researchwork on WOMs, e.g., [5], [22], [31], [34], and [36].Moreover, usually in programming of the cells, we let the

charge level in a single cell of a flash memory only to be raised,and hence, the errors in a single cell will be asymmetric. Asym-metric error-correcting codes were subject to extensive researchdue to their applications in coding for computer memories [18].The errors in a cell of a flash memory are a new type of asym-metric errors that have limited magnitude. Errors in this modelare in one direction and are not likely to exceed a certain limit.This means that a cell in level can be raised by an error tolevel , such that and , whereis the error limited magnitude. Asymmetric error-correctingcodes with limited magnitude were proposed in [1] and werefirst considered for nonvolatile memories in [3] and [4]. Re-cently, several other papers have considered the problem, e.g.,[7], [8], [13], and [35].In this work, we will consider a solution for both the

construction problem of asymmetric codes with limitedmagnitude and the coding problem in WOMs. Our proposedsolution will use an older concept in combinatorics namedtiling. Given an -dimensional shape , a tiling of

with consists of disjoint copies of such that eachpoint of is covered by exactly one copy of . Tiling is awell-established concept in combinatorics and especially incombinatorial geometry. There are many algebraic methodsrelated to tiling [28] and it is an important topic also in codingtheory. For example, perfect codes are associated with tilings,where the related sphere is the -dimensional shape . Tiling isdone with a shape and we consider only shapes that form anerror sphere for asymmetric limited-magnitude codes or theirimmediate generalization in . The definition of a tiling inwill be given in Section II.Two of the most considered shapes for tiling are the cross

and the semicross [26], [28]. These were also considered in con-nections to flash memories [21]. In this paper, we will consideranother shape which will be called in the sequel an -dimen-sional chair. An -dimensional chair is an -dimensional boxfrom which a smaller -dimensional box is removed from oneof its corners. This is a generalization of the original conceptwhich is an -cube from which one vertex was removed [16].In other places, this shape is called a notched cube [15], [20],[27]. Lattice tiling with this shape will be discussed, regardlessof the length of each side of the larger box and the length ofeach side of the smaller box. We will show an equivalent way

0018-9448/$31.00 © 2012 IEEE

1574 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 3, MARCH 2013

to present a lattice tiling. This method will be called a general-ized splitting and it generalizes the concepts of splitting definedin [23]; and the concept of sequences defined and used forthe construction of codes correcting asymmetric errors with lim-ited magnitude in [13]. We will show two applications of tilingswith such a shape. One application is for the construction ofcodes that correct up to asymmetric limited-magnitudeerrors with any given magnitude for each cell; and a second ap-plication is for constructing WOM codes with multiple writing.In the first part of this work, we will consider only tilings with-dimensional chairs. In the second part of this work, we willconsider the applications of tilings with -dimensional chairs.The rest of this paper is organized as follows. In Section II, wedefine the basic concepts for our presentation of tilings with-dimensional chairs. We define the concepts of an -dimen-sional chair and a tiling of the space with a given shape. Wepresent the -dimensional chair as a shape in . When the-dimensional chair consists of unit cubes connected only byunit cubes of smaller dimensions, the -dimensional chair canbe represented as a shape in . For such a shape, we will seekfor an integer tiling. We will be interested in this paper only inlattice tiling and when the shape is in only in integer lat-tice tiling. Two representations for tiling with a shape will begiven. The first representation is with a generator matrix for thelattice tiling and the second is by the concept which is called ageneralized splitting. We will show that these two representa-tions are equivalent. In Section III, we will present a construc-tion for tilings with -dimensional chairs based on generalizedsplitting. The construction will be based on properties of someAbelian groups. In Section IV, we will present a constructionof tiling with -dimensional chairs based on lattices. This con-struction works on any -dimensional chair, while the construc-tion of Section III works only on certain discrete ones. We notethat after the paper was written, it was brought to our attentionthat lattice tilings for notched cubes were given in [15], [20],and [27]. For completeness and since our proof is slightly dif-ferent, we kept this part in this paper. Tiling with a discrete -di-mensional chair can be viewed as a perfect code for correctionof asymmetric errors with limited magnitude. In Section V, wepresent the definition for such codes, not necessarily perfect. Wealso present the necessary definition for such perfect codes. Weexplain what kind of perfect codes are derived from our con-structions and also how non-perfect codes can be derived fromour constructions. In Section VI, we prove that certain perfectcodes for correction of asymmetric errors with limited magni-tude do not exist. In Section VII, we will discuss the applicationof our construction for multiple writing in cells -ary WOM.We conclude this paper in Section VIII.

II. BASIC CONCEPTS

An -dimensional chair , ,, for each , ,

is an -dimensional box from which an-dimensional box was removed from oneof its corners. Formally, it is defined by

Fig. 1. Semicross with and a 3-D chair with and.

For a given -dimensional shape , let denote the volumeof . The following lemma on the volume of is an imme-diate consequence of the definition.

Lemma 1: If andare two vectors in , where for each , ,then

If ,then the -dimensional chair is a discrete shape and itcan be viewed as a collection of connected -dimensional unitcubes in which any two adjacent cubes share a complete-dimensional unit cube. In this case, the formal definition of

the -dimensional chair, which considers only points of , is

Remark 1: It is important to note that ifand

are two integer vectors, then the two definitions coincide onlyif is viewed as a collection of -dimensional unit cubes.Special consideration, in the definition, should be given to theboundaries of the cubes, but this is not an issue for this work.For , if and , then the chair

coincides with the shape known as a corner (or a semicross)[25]. Examples of a 2-D semicross and a 3-D chair are given inFig. 1.A set is a packing of with a shape if copies ofplaced on the points of (in the same relative position of )

are disjoint. A set is a tiling of with a shape if it isa packing and the disjoint copies of in the packing cover .A set is a packing of with a shape if copies ofplaced on the points of (in the same relative position of )

have nonintersecting interiors. The closure of a shapeis the union of with its exterior surface. A set is atiling of with a shape if it is a packing and the closure, ofthe distinct copies of in the packing, covers .In the rest of this section, we will describe two methods to

represent a packing (tiling) with a shape . The first representa-tion is with a lattice. In case that is a discrete shape, we havea second representation with a splitting sequence.A lattice is an additive subgroup of . We will assume

that

BUZAGLO AND ETZION: TILINGS WITH -DIMENSIONAL CHAIRS AND THEIR APPLICATIONS TO ASYMMETRIC CODES 1575

where is a set of linearly independent vectorsin . The set of vectors is called the basis for, and the matrix

......

. . ....

having these vectors as its rows is said to be the generator matrixfor . If , then the lattice is called an integer lattice.The volume of a lattice , denoted by , is inversely pro-

portional to the number of lattice points per a unit volume. Thereis a simple expression for the volume of , namely,

.A lattice is a lattice packing (tiling) with a shape if the

set of points of forms a packing (tiling) with . The followinglemma is well known.

Lemma 2: A necessary condition that a lattice defines alattice packing (tiling) with a shape is that( ). A sufficient condition that a lattice packingdefines a lattice tiling with a shape is that .In the sequel, let denote the unit vector with an one in theth coordinate, let denote the all-zero vector, and let de-note the all-one vector. For two vectors ,

, and a scalar , we define the

vector addition ,

and the scalar multiplication . Fora set and a vector , the shift of by is

.Let be an Abelian group and let

be a sequence with elements of . For every, we define

where addition and multiplication are performed in .A set splits an Abelian group with a splitting se-

quence , , for each , ,if the set contains distinct elements from. We will call this operation a generalized splitting. The split-

ting defined in [11] and discussed in [12], [23], [24], and [26]is a special case of the generalized splitting. It was used for theshapes known as cross and semicross [24], [25], and quasi-cross[21]. The sequences defined in [13] and discussed in [13]and [14] for the construction of codes that correct asymmetricerrors with limited magnitude are also a special case of the gen-eralized splitting. These sequences are modification of thewell-known Sidon sequences and their generalizations [2]. Thegeneralized splitting also makes generalization for a methoddiscussed by Varshamov [29], [30]. The generalization can beeasily obtained, but to our knowledge a general and completeproven theory was not given earlier.

Lemma 3: If is a lattice packing of with a shape ,then there exists an Abelian group of order , such thatsplits .

Proof: Let and let be the grouphomomorphism that maps each element to the coset

. Clearly, .Let be a sequence defined by

for each , . Clearly, for each , we have.

Now, assume that there exist two distinct elements, such that

It implies that

Since if and only if , it follows that thereexists , , such that

Therefore, , which contradicts the fact thatis a lattice packing of with the shape .Thus, splits with the splitting sequence .

Lemma 4: Let be an Abelian group and let be a shape in. If splits with a splitting sequence , then there exists a

lattice packing of with the shape , for which .Proof: Consider the group homomorphism

defined by

Clearly, is a lattice and the volume of ,.

To complete the proof, we have to show that is a packingof with the shape . Assume to the contrary that there ex-ists such that . Hence, there existtwo distinct elements such that andtherefore

Therefore, , which contradicts the fact that splitswith the splitting sequence .Thus, is a lattice packing with the shape .

Corollary 1: A lattice tiling of with the shapeexists if and only if there exists an Abelian group of ordersuch that splits .If our shape is not discrete, i.e., cannot be represented

by a set of -dimensional units cubes, two of which are adja-cent only if they share an -dimensional unit cube, thenclearly tiling can be represented with a lattice, but cannot berepresented with a splitting sequence. But, if our shape isin , then we can use both methods as they were proved tobe equivalent. In fact, both methods are complementary. If weconsider the matrix , then the vector

is contained in the related latticeif and only if . Therefore, has some similarity to


a parity-check matrix in coding theory. The representation of alattice with its generator matrix seems to be more practical. But,sometimes it is not easy to construct one. Moreover, the split-ting sequence has in many cases a nice structure and from itsstructure, the general structure of the lattice can be found. Thisis the case in the next two sections. In Section III, we presenttwo constructions of tilings based on generalized splitting. Eventhough the second one generalizes the first one, the mathemat-ical structure of the first one has its own beauty, and hence, bothconstructions are given. The construction of the lattice, in ,given in Section IV, was derived based on the structure of thelattices, in , obtained from the construction of the splittingsequences in Section III.

III. CONSTRUCTIONS BASED ON GENERALIZED SPLITTING

In this section, we will present a construction of a tiling with-dimensional chairs based on generalized splitting. The -di-mensional chairs that are considered in this section are discrete,i.e., . We start with a construction in which all the’s are equal to , and all the ’s are equal to . We gen-eralize this construction to a case in which all the ’s, with apossible exception of one, have multiplicative inverses in therelated Abelian group.For the ring , the ring of integers modulo , let be

the multiplicative group of formed from all the elements ofthat have multiplicative inverses in .

Lemma 5: Let , , be two integers and let be thering of integers modulo , i.e., . Then

(P1) and are elements of .(P2) is an element of order in .(P3) equals to zero in .Proof:(P1) By definition, is zero in .We also have that

. It follows thatin , and hence,

. Since is zero in , it followsthat , and hence, if and only if

.(P2) Clearly, and since ,it follows that . This also im-plies that has a multiplicative inverse, and hence,

.Now, note that for each , , wehave . Therefore,

in , and hence, .Thus, the order of in is .(P3) Clearly,

. By definition, , and hence,, , , , and

. Therefore,, which implies that .

Theorem 6: Let , , be two integers,, and . Then, ,

, , splitswith the splitting sequence defined by

Proof: We will show by induction that every element incan be expressed in the form , for some .The basis of induction is .For the induction step, we have to show that if

can be presented as for some (i.e.,, , ,

and for some , ), then also can be presentedin the same way. In other words, , where

.If and there exists such that , then

where , , , and the inductionstep is proved.If and there is no such that , then by

Lemma 5(P3), we have that and hence

i.e., is the required element of and theinduction step is proved.Now, assume that . Let , be the smallest

index such that

Note that for each ,

Therefore

If , then and the induction stepis proved. If , i.e., , then

By iteratively continuing in the same manner, we obtain

and since , we have that

and the induction step is proved.


Since , it follows that the sethas elements.

Corollary 2: For each and , there ex-ists a lattice tiling of with , ,

.The next theorem and its proof are generalizations of The-

orem 6 and its proof.

Theorem 7: Let andbe two vectors in such that for

each , . Let , , ,and assume that for each , , . Then,splits with the splitting sequence definedby

Proof: First, we will show that . Sinceequals zero in , it follows that in , and hence,

. Therefore

As an immediate consequence from definition, we have that foreach ,

Next, we will show that

(1)

Since , it follows that to prove Theorem 7, it issufficient to show that each element in can be expressed as

, for some . The proof will be done by induction.The basis of induction is .In the induction step, we will show that if can be pre-

sented as for some , then the same is true for .In other words, , where

.Assume

where , for each , and thereexists a such that .If or if and there exists

such that , then since , it follows that

where . Clearly, ;if and otherwise . Hence, theinduction step is proved.If and there is no such that ,

then by (1), we have that and hence

i.e., is the required element of andthe induction step is proved.Now, assume that . Let be the smallest

index such that

If , then and the inductionstep is proved. If , then

By iteratively continuing in the same manner, we obtain

and since , we have that

is the element of , and the induction step is proved.

Corollary 3: Let andbe two vectors in such that

for each , . Let and assume thatfor at least of the ’s. Then, there exists

a lattice tiling of with .

IV. TILING BASED ON A LATTICE

Next, we consider lattice tiling of with ,where and .We want to remind again that Mihalis Kolountzakis and JamesSchmerl pointed on [15], [20], and [27], where such tiling canbe found. For completeness and since our proof is slightly dif-ferent, we kept this part in this paper. For the proof of the nexttheorem, we need the following lemma.

Lemma 8: Let . Then,if and only if , for ,

and there exist integers and , , such thatand .

Proof: Assume first that , i.e., thereexists . By the definitionof , it follows that

(2)


and there exists a such that

(3)

Similarly, for , we have

(4)

and there exists an such that

(5)

It follow from (2) and (4) that andfor each , . Hence,

for each , . It follow from (3) and (4) that. It follows from (5) and (2) that

.Now, let such that for

each , , and there exist such that and. Consider the point

, where .By definition, for each ,

and . Hence, .Clearly, if , then and if , then .

In both cases, since , it follows that we have

We also have , and therefore,. Hence, .

Thus, , i.e., .

The next theorem is a generalization of Corollary 3.

Theorem 9: Let and, , for all .

Let be the lattice generated by the matrix

......

. . .. . .

. . ....

Then, is a lattice tiling of with .Proof: It is easy to verify that

. We will use Lemma 2 to showthat is a tiling of with . For this, it is sufficient toshow that is a packing of with .Let , , and assume to the contrary that

. Since , it follows that thereexist integers , not all zeros, such that

, for every , . By Lemma 8, wehave that for each ,

i.e.,

Since is an integer, it follows that or

. For each , , if , then since, we have that

Hence

(6)

Similarly, if , we have that

If , then by (6), we have

and hence for each , . Similarly, we havefor each , if . If , then since

is an integer, we have that , foreach , . Hence, there is no such that ,which contradicts Lemma 8. Similarly, if , then for each, , , and hence, thereis no such that , which contradicts Lemma8. Therefore, , i.e., for each , , , acontradiction. Hence, is a lattice packing of withThus, by Lemma 2, is a lattice tiling of with .

Remark 2: Note that the construction (Theorem 9) based onlattices covers all the parameters of integers that are not coveredin Section III.

V. ASYMMETRIC ERRORS WITH LIMITED MAGNITUDE

The first application for a tiling of with an -dimensionalchair is in the construction of codes of length that correctasymmetric errors with limited magnitude.Let be an alphabet with letters. For a

word , the Hamming weight of ,, is the number of nonzero entries in , i.e.,

.A code of length over the alphabet is a subset of .

A vector is a -asymmetric-error with lim-ited magnitude if and for each

. The sphere is the set of all -asymmetric errorswith limited magnitude . A code can correct -asym-metric-errors with limited magnitude if for any two codewords

, and any two -asymmetric-errors with limited magni-tude , , such that , , wehave that .The size of the sphere is easily computed.

Lemma 10: .

Corollary 4: .


For simplicity, it is more convenient to consider the code asa subset of , where all the additions are performed modulo .Such a code can be viewed also as a subset of formed bythe set . This code is an extension,from to , of the code . Note, in this code, there is a wraparound (of the alphabet) which does not exist if the alphabet is, as in the previous code.A linear code , over , which corrects -asymmetric-errors

with limited magnitude , viewed as a subset of , is equivalentto an integer lattice packing of with the shape .Therefore, we will call this lattice a lattice code.Let denote the set of lattice codes in that cor-

rect -asymmetric-errors with limited magnitude . A codeis called perfect if it forms a lattice tiling with the

shape . By Corollary 1, we have the following.

Corollary 5: A perfect lattice code exists ifand only if there exists an Abelian group of ordersuch that splits .A code is formed as an extension of a code

over . Assume we want to form a code , where

, which corrects asymmetric errorswith limited magnitude . Assume that a construction with alarge linear code does not exist. One can take a latticecode over an alphabet with letters . Then,

a code over the alphabet is formed by . Note thatthe code is usually not linear. This is a simple constructionthat always works. Of course, we expect that there will be manyalphabets in which better constructions can be found.There exists a perfect lattice code for various

parameters with [13], [14]. Such codes also exist forand all and for the parameters of the Golay codes and thebinary repetition codes of odd length [17].The existence of perfect codes that correct ( )-asym-

metric-errors with limited magnitude was proved in [14]. Therelated sphere is an -dimensional chair ,where and .Sections III and IV provide constructions for such codes withsimpler description and simpler proofs that these codes are suchperfect codes.In fact, the constructions in these sections provide tilings of

many other related shapes. More than that, there might be sce-narios in which different flash cells can have different limitedmagnitude, e.g., if for some cells wewant to increase the numberof charge levels. In this case, we might need a code that cor-rect asymmetric errors with different limited magnitudes for dif-ferent cells. Assume that for the th cell the limited magnitudeis . Our lattice tiling with ,

, , produces the required perfectcode for this scenario.

VI. NONEXISTENCE OF SOME PERFECT CODES

Next, we ask whether perfect codes, which correct asym-metric errors with limited magnitude, exist for . Un-fortunately, such codes cannot exist. The proof for this claim isthe goal of this section. Most of the proof is devoted to the casein which the limited magnitude is equal to one. We concludethe section with a proof for .

For a word , we define

We say that a codeword , , covers a wordif there exists an element such that.

Lemma 11: Let , and assume that there exists, , such that , for

every , . Then, or .Proof: Let , , such that, for every , . Assume to the contrary that

and . Letwhere and where

. Clearly, and.

Therefore, , which contradictsthe fact that . Thus, or

.

Lemma 12: Let be a lattice code. Theword, the all-one vector, can be covered only by a codeword

of the form , for some , , where is aninteger, .

Proof: Assume that is the codeword that covers .Then, there exists suchthat , i.e., , and therefore,for each , . Since , it follows that thereare at least two entries that are equal to one in . By Lemma 11,it follows that . Hence, there are at leastentries of which are equal to one. Therefore,for some , , where is an integer, .

Lemma 13: Let be a lattice code. Forevery , , the word can be coveredonly by a codeword of the form , where is an integer,

.Proof: Assume that is a codeword that covers .

Then, there exists suchthat . Clearly, , and for each

, and therefore for each ,. Since , it follows that there are at

most negative coordinates in . Therefore, by Lemma 11,it follows that . Hence, there are at leastcoordinates of that are equal to one. Thus, ,where .

Lemma 14: If there exists a perfect lattice codein , then divides

for some integer ,.

Proof: Let be a perfect lattice code. ByLemma 12 and w.l.o.g., we can assume that is covered by thecodeword , where . Combining this withLemma 13, we deduce that for all , , the word

is covered by the codeword( cannot be equal , since it would coverwhich is already covered by ). We have distinct codewordsin , and since is a lattice, the lattice generated by the


set is a sublattice of , and therefore,divides . Let be the matrix

whose rows are

. . ....

.... . .

. . ....

...

Subtracting the first row from every other row, we obtain thedeterminant

. . ....

.... . .

. . ....

...

Subtracting the first column from all the other columns, exceptfrom the last one, we obtain the determinant

. . ....

.... . .

. . ....

...

Finally, replacing the second row by the sum of all the rows,except for the first one, we obtain the determinant

. . ....

.... . .

. . ....

...

Now, it is easy to verify that

.

Theorem 15: There are no perfect lattice codes infor all .

Proof: By Lemma 14, it is sufficient to showthat does not divide

, for .If , then we have to show that does not

divide . It can be readily verified thatfor all , which proves the claim.If , then we have to show that does not

divide . If , then. Hence, we have to show that

does not divide . We will show that for all ,. It is easy to verify that

Therefore, it is sufficient to show that

or equivalently

This is simply proved by induction on for all .To complete the proof, we should only verify that for ,

5, and 6, we have that does not divide .

Theorem 16: There are no perfect lattice codes inif and .

Proof: Let and and assume to the contrary thatthere exists a perfect lattice code . Withoutloss of generality (w.l.o.g.), we can assume by Lemma 12 thatthe word is covered by a codeword ,where is an integer, . From the proof of Lemma 14,we have that for all , , the wordis covered by the codeword . Therefore,

is a codeword. Clearly, and since, it follows that for all , , . Moreover,

and , which contradictsLemma 11. Thus, if and , then there are no perfectlattice codes in .

Combining Theorems 15 and 16, we obtain the main result ofthis section.

Corollary 6: There are no perfect lattice codes inif for any limited magnitude .

The existence of perfect lattice codes in andtheir nonexistence in might give an evidence thatsuch perfect codes would not exist in forand some . It would be interesting to prove such a claimfor and .

VII. APPLICATION TO WOMS

A second possible application for a tiling of with an -di-mensional chair is in constructions of multiple writing in cellsWOMs. Each cell has charge levels . A letterfrom an alphabet of size , , is written intothe cells as many times as possible. In each round, the chargelevel in each cell is greater than or equal to the charge level inthe previous round. It is desired that the number of rounds forwhich we can guarantee to write a new symbol from will bemaximized.An optimal solution for the problem can be described as fol-

lows. Let be an -dimensional array. Letbe a coloring of the array with the alphabet

letters. The rounds of writing and raising the charge levels of


the cells can be described in terms of the coloring of thearray . If in the first round, the symbol is written and thecharge level in cell is raised to , , then the colorin position is . Therefore, we have to find acoloring function such that the number of rounds in which anew symbol can be written will be maximal.Cassuto and Yaakobi [5] have found that using a coloring

based on a lattice tiling with a 2-D chair provides the bestknown writing strategy when there are two cells. A coloringof based on a lattice tiling with a shape has

colors. The lattice have cosets, and hence coset repre-sentatives, . The points in of the coset

are colored with the th letter of . Now, the coloring ofentry of given by is equal to the color ofthe point given by the coloring . Themethod given in [5] suggests that a generalization using col-oring based on tiling of with an -dimensional chair will bea good strategy for WOM codes with cells [33]. The analysiswith two cells, i.e., 2-D tiling, was discussed with more detailsin [5]. The analysis for the -dimensional case will be discussedin a future research work by the same authors and another groupas well [33].

VIII. CONCLUSION

We have presented a few constructions for tilings with -di-mensional chairs. The tilings are based either on lattices or ongeneralized splitting. Both methods are equivalent if our spaceis . The generalized splitting is a simple generalization forknown concepts such as splitting and sequences. We haveshown that our tilings can be applied in the design of codesthat correct asymmetric errors with limited magnitude. We fur-ther mentioned a possible application in the design of WOMcodes for multiple writing. Finally, we proved that some perfectcodes for correction of asymmetric errors with limited magni-tude cannot exist.

ACKNOWLEDGMENT

The authors would like to thank Mihalis Kolountzakis andJames Schmerl for pointing on references [15], [20], [27], whichconsider lattice tiling of notched cubes.

REFERENCES

[1] R. Ahlswede, H. Aydinian, and L. Khachatrian, “Unidirectional errorcontrol codes and related combinatorial problems,” in Proc. 8th Int.Workshop Algebr. Combin. Coding Theory, St. Petersburg, Russia,2002, pp. 6–9.

[2] K. O’Bryant, “A complete annotated bibliography of work related toSidon sequences,”Electron. J. Combin., vol. DS11, pp. 1–39, Jul. 2004.

[3] Y. Cassuto, M. Schwartz, V. Bohossian, and J. Bruck, “Codesmulti-level flash memories: Correcting asymmetric limited-magnitudeerrors,” in Proc. IEEE Int. Symp. Inf. Theory, Nice, France, 2007, pp.1176–1180.

[4] Y. Cassuto, M. Schwartz, V. Bohossian, and J. Bruck, “Codes forasymmetric limited-magnitude errors with application to multi-levelflash memories,” IEEE Trans. Inf. Theory, vol. IT-56, no. 4, pp.1582–1595, Apr. 2010.

[5] Y. Cassuto and E. Yaakobi, “Short -ary WOM codes with hot/coldwrite differentiation,” in Proc. IEEE Int. Symp. Inf. Theory, Boston,MA, 2012, pp. 1396–1400.

[6] G. D. Cohen, P. Godlewski, and F. Merkx, “Linear binary code forwrite-once memories,” IEEE Trans. Inf. Theory, vol. IT-32, no. 5, pp.697–700, Sep. 1986.

[7] L. Dolecek, “Towards longer lifetime of emerging technologies usingnumber theory,” in Proc. Workshop Appl. Commun. Theory Emerg.Memory Technol., Miami, FL, 2010, pp. 1936–1940.

[8] N. Elarief and B. Bose, “Optimal, systematic, -ary codes correcting allasymmetric and symmetric errors of limited magnitude,” IEEE Trans.Inf. Theory, vol. IT-56, no. 3, pp. 979–983, Mar. 2010.

[9] A. Fiat and A. Shamir, “Generalized write-once memories,” IEEETrans. Inf. Theory, vol. IT-30, no. 3, pp. 470–480, May 1984.

[10] F. Fu and H. J. Han Vinck, “On the capacity of generalized write oncememory with state transitions described by an arbitrary directed acyclicgraph,” IEEE Trans. Inf. Theory, vol. IT-45, no. 1, pp. 308–313, Jan.1999.

[11] G. Hajós, “Uber einfache und mehrfache Bedeckung des -dimen-sionalen Raumes mit einem Würfelgitter,” Math. Zeit., vol. 47, pp.427–467, 1942.

[12] D. Hickerson, “Splitting of finite groups,” Pacific J. Math., vol. 107,pp. 141–171, 1983.

[13] T. Kløve, B. Bose, and N. Elarief, “Systematic single limited magni-tude asymmetric error correcting codes,” IEEE Trans. Inf. Theory, vol.IT-57, no. 7, pp. 4477–4487, Jul. 2011.

[14] T. Kløve, J. Luo, I. Naydenova, and S. Yari, “Some codes correctingasymmetric errors of limited magnitude,” IEEE Trans. Inf. Theory, vol.IT-57, no. 11, pp. 7459–7472, Nov. 2011.

[15] M. Kolountzakis, “Lattice tilings by cubes: Whole, notched and ex-tended,” Electron. J. Combin., vol. 5, pp. 1–11, 1998.

[16] J.-Y. Lee and R. V. Moody, “Lattice substitution system and modelsets,” Discr. Comput. Geom., vol. 25, pp. 173–201, 2001.

[17] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-CorrectingCodes. Amsterdam, The Netherlands: North-Holland, 1977.

[18] T. R. N. Rao and E. Fujiwara, Error-Control Coding for ComputerSystems. Englewood Cliffs, NJ: Prentice-Hall, 1989.

[19] R. L. Rivest and A. Shamir, “How to reuse a write-once memory,” Inf.Control, vol. 55, pp. 1–19, 1982.

[20] J. H. Schmerl, “Tiling space with notched cubes,” Discr. Math., vol.133, pp. 225–235, 1994.

[21] M. Schwartz, “Quasi-cross lattice tilings with applications to flashmemory,” in Proc. IEEE Int. Symp. Inf. Theory, St. Petersburg, Russia,2011, pp. 2133–2137.

[22] A. Shpilka, New constructions of WOM codes using the Wozencraftensemble [Online]. Available: arxiv.org/abs/1110.6590

[23] S. Stein, “Factoring by subsets,” Pacific J. Math., vol. 22, pp. 523–541,1967.

[24] S. Stein, “Algebraic tiling,” Amer. Math. Mon., vol. 81, pp. 445–462,1967.

[25] S. Stein, “Packing of by certain error spheres,” IEEE Trans. Inf.Theory, vol. IT-30, no. 2, pp. 356–363, Mar. 1984.

[26] S. Stein, “Tiling, packing, and covering by clusters,” Rocky MountainJ. Math., vol. 16, pp. 277–321, 1986.

[27] S. Stein, “The notched cube tiles ,” Discr. Math., vol. 80, pp.335–337, 1990.

[28] S. Stein and S. Szabó, Algebra and Tiling. Cambridge, U.K.: Cam-bridge Univ. Press, 1994.

[29] R. R. Varshamov, “On some specifics error-correcting linear codes,”Rep. Acad. Sci. USSR, vol. 157, pp. 546–548, 1964.

[30] R. R. Varshamov, “On the theory of asymmetric codes,” Rep. Acad.Sci. USSR, vol. 164, pp. 757–760, 1965.

[31] L. Wang, M. Qin, E. Yaakobi, Y.-H. Kim, and P. H. Siegel, “WOMwith retained messages,” in Proc. IEEE Int. Symp. Inf. Theory, Boston,MA, 2012, pp. 1396–1400.

[32] J. K. Wolf, A. D. Wyner, J. Ziv, and J. Korner, “Coding for write-oncememory,” Bell Labs. Tech. J., vol. 63, pp. 1089–1112, 1984.

[33] E. Yaakobi, Personal communication 2012.[34] E. Yaakobi and A. Shpilka, “High sum-rate three-write and non-binary

WOM codes,” inProc. IEEE Int. Symp. Inf. Theory, Boston,MA, 2012,pp. 1391–1395.

[35] E. Yaakobi, P. H. Siegel, A. Vardy, and J. K. Wolf, “On codes thatcorrect asymmetric errors with graded magnitude distribution,” inProc. IEEE Int. Symp. Inf. Theory, St. Petersburg, Russia, 2011, pp.1021–1025.

[36] E. Yaakobi, P. H. Siegel, A. Vardy, and J. K. Wolf, “Multiple error-correcting WOM-codes,” IEEE Trans. Inf. Theory, vol. IT-58, no. 4,pp. 2220–2230, Apr. 2012.

[37] Z. Zemor and G. D. Cohen, “Error-correcting WOM-codes,” IEEETrans. Inf. Theory, vol. IT-37, no. 3, pp. 730–734, May 1991.


Sarit Buzaglo was born in Israel in 1983. She received the B.A. and M.Sc. de-grees from the Technion—Israel Institute of Technology, Haifa, Israel, in 2007and 2010, respectively, from the department of Mathematics. She is currentlya Ph.D. student in the Computer Science Department at the Technion. Her re-search interests include algebraic error-correction coding, coding theory, dis-crete geometry, and combinatorics.

Tuvi Etzion (M’89–SM’94–F’04) was born in Tel Aviv, Israel, in 1956. Hereceived the B.A., M.Sc., and D.Sc. degrees from the Technion—Israel Instituteof Technology, Haifa, Israel, in 1980, 1982, and 1984, respectively.From 1984 he held a position in the department of Computer Science at

the Technion, where he has a Professor position. During the years 1986-1987he was Visiting Research Professor with the Department of Electrical En-gineering—Systems at the University of Southern California, Los Angeles.During the summers of 1990 and 1991 he was visiting Bellcore in Morristown,New Jersey. During the years 1994–1996 he was a Visiting Research Fellowin the Computer Science Department at Royal Holloway College, Egham,England. He also had several visits to the Coordinated Science Laboratoryat University of Illinois in Urbana-Champaign during the years 1995–1998,two visits to HP Bristol during the summers of 1996, 2000, a few visits to thedepartment of Electrical Engineering, University of California at San Diegoduring the years 2000–2012, and several visits to the Mathematics departmentat Royal Holloway College, Egham, England, during the years 2007–2009.His research interests include applications of discrete mathematics to prob-

lems in computer science and information theory, coding theory, and combina-torial designs.Dr. Etzion was an Associate Editor for Coding Theory for the IEEE

TRANSACTIONS ON INFORMATION THEORY from 2006 till 2009.

ieee transections on information theory - technion

Documents