product constructions for perfect lee codes

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 11, NOVEMBER 2011 7473

Product Constructions for Perfect Lee CodesTuvi Etzion, Fellow, IEEE

Abstract—Awell-known conjecture of Golomb andWelch is thatthe only nontrivial perfect codes in the Lee and Manhattan met-rics have length two or minimum distance three. This problemand related topics were subject for extensive research in the last40 years. In this paper, two product constructions for perfect Leecodes and diameter perfect Lee codes are presented. These con-structions yield a large number of nonlinear perfect codes and non-linear diameter perfect codes in the Lee and Manhattan metrics.A short survey and other related problems on perfect codes in theLee and Manhattan metrics are also discussed.

Index Terms—Anticode, diameter perfect code, Hammingscheme, Lee metric, Manhattan metric, perfect code, periodiccode, product construction.

I. INTRODUCTION

T HE Lee metric was introduced in [1] and [2]for transmission of signals taken from

over certain noisy channels. It was generalized forin [3]. The Lee distance between two words

isgiven by .A related metric, the Manhattan metric, is defined foralphabet letters taken as any integer. For two words

,the Manhattan distance between and is defined as

. A code in either metric(and in any other metric as well) has minimum distance if foreach two distinct codewords we have ,where stands for either the Lee distance or the Manhattandistance (or any other distance measure). Linear codes are usu-ally the codes which can be handled more effectively and hencelinear codes will be the building blocks in our constructions.We will not restrict ourself only for linear codes, but we willalways assume that the all-zero word is a codeword.A linear code in is an integer lattice. A lattice is a dis-

crete, additive subgroup of the real -space . Without loss ofgenerality (w.l.o.g.), we can assume that

(1)

where is a set of linearly independent vectorsin . A lattice defined by (1) is a sublattice of if andonly if . We will be interested solely in

Manuscript received March 21, 2011; accepted June 16, 2011. Date of cur-rent version November 11, 2011. This work was supported in part by the IsraeliScience Foundation (ISF), Jerusalem, Israel, under Grant 230/08.The author is with the Department of Computer Science, Technion—Israel

Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]).Communicated by G. Cohen, Associate Editor for Coding Theory.Digital Object Identifier 10.1109/TIT.2011.2161133

sublattices of . The vectors are called the basisfor , and the matrix

......

. . ....

having these vectors as its rows is said to be the generator matrixfor . The lattice with the generator matrix is denoted by

.

Remark 1: There are other ways to describe a linear code inthe Manhattan metric and the Lee metric. The traditional wayof using a parity-check matrix can be also used [4], [5]. But, inour discussion, the lattice representation is the most convenient.

The volume of a lattice , denoted , is inversely propor-tional to the number of lattice points per unit volume. More pre-cisely, may be defined as the volume of the fundamentalparallelogram , which is given by

There is a simple expression for the volume of , namely,.

A shape tiles if disjoint copies of cover all the pointsof . The cover of with disjoint copies of is called atiling. We say that induces a lattice tiling of a shape if dis-joint copies of placed on the lattice points on a given specificpoint in form a tiling of .Codes in generated by a lattice are periodic. We say that

the code has period if for each ,, the word is a codeword if and

only if . Let be theleast common multiplier of the period . Thecode has also period and the code can bereduced to a code in the Lee metric over the alphabet withthe same minimum distance as . The parameters of such codewill be given by , where is the length of the code,is its minimum distance, is the volume of the related lattice

( is the redundancy of the code), and is the alphabetsize. The number of codewords in such a code is .The research on codes with theManhattan metric is not exten-

sive. It is mostly concern with the existence and nonexistence ofperfect codes [3], [6]–[8]. Nevertheless, all codes defined in theLee metric over some finite alphabet can be extended to codes inthe Manhattan metric over the integers. The code resulting froma Lee code over is periodic with period . Theminimum Manhattan distance will be the same as the minimumLee distance. The literature on codes in the Lee metric is veryextensive, e.g., [4] and [9]–[16]. The interest in Lee codes hasbeen increased in the last decade due to many new and diverse

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7474 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 11, NOVEMBER 2011

applications of these codes. Some examples are constrained andpartial-response channels [4], interleaving schemes [17], multi-dimensional burst error correction [18], and error correction forflash memories [19]. The increased interest is also due to newattempts to settle the existence question of perfect codes in thesemetrics [8].Perfect codes is one of the most fascinating topics in coding

theory. A perfect code in a given metric is a code in which theset of spheres with a given radius around its codewords forma partition of the space. These codes were mainly considered forthe Hamming scheme, e.g., [20]–[26]. They were also consid-ered for other schemes as the Johnson scheme and the Grass-mann scheme, But, as said, these codes were considered to alarger extent also in the Lee and Manhattan metrics.This paper wasmotivated by some basic concepts which were

presented in [27] and [28]. There are two goals for this research.The first one is to present product constructions for perfect codesin the Lee and Manhattan metrics in a similar way to what wasdone in the Hamming scheme. The second one is to show howthese constructions can be used to solve other problems relatedto perfect codes in the Lee andManhattan metrics. One problemis the number of different perfect codes and diameter perfectcodes in the Lee and Manhattan metrics. A second problem isthe existence of non-periodic perfect codes in the Manhattanmetric. In the process, we will also present a new product con-struction for perfect codes in the Hamming scheme.The rest of this paper is organized as follows. In Section II,

we introduce the basic concepts which are used in our paper.We first define a perfect code in general and discuss some of theknown results on perfect codes and extended perfect codes inthe Hamming scheme and the known results on perfect codesin the Lee and Manhattan metrics. We continue by introducingthe concept of anticodes and diameter perfect code as was firstdiscussed in [29]. We prove that the definition of diameter per-fect codes can be applied to codes in the Lee and Manhattanmetrics. We discuss the knowledge on these codes. We presenta simple construction for diameter perfect codes with minimumdistance four in the Lee and Manhattan metrics. Finally, we de-scribe a doubling construction for perfect codes in the Ham-ming scheme which was given by Phelps [20]. In Section III,we present a new product construction for -ary perfect codesin the Hamming scheme. Similar idea to the one in this construc-tion was used in other papers, e.g., [24] and [30]–[32], for con-struction of relatively large binary codes (perfect and nonper-fect) with minimum Hamming distance three in the Hammingscheme. In Section IV, we present two product constructions,one for perfect codes and one for diameter perfect codes in theLee and Manhattan metrics. These constructions are modifica-tions of the two product constructions in the Hamming scheme.In Section V, we discuss two problems related to perfect codesand diameter perfect codes in the Lee and Manhattan metrics.The first one is the number of nonequivalent such codes and thesecond one is the existence of nonperiodic perfect codes andnonperiodic diameter perfect codes. We will show how our con-structions can be used in the context of these two problems. Asummary and a list of questions for future research are given inSection VI.

II. BASIC CONCEPTS AND CONSTRUCTIONS

A. Spheres and Perfect Codes

The main two concepts in this paper are perfect codes andcodes in the Lee and Manhattan metrics. We start with a generaldefinition of perfect codes. For a given space , with a distancemeasure , a subset of is a perfect code with radiusif for every element there exists a unique codeword

such that . For a point , the sphereof radius around , , is the set of elements in suchthat if and only if , i.e.,

. For the sphere , is calledthe center of the sphere. In this paper, we consider only theHamming scheme and the Lee and Manhattan metrics, in whichthe size of a sphere does not depend on the center of the sphere.Hence, in the rest of the paper, we will assume that in our metricthe size of a sphere with radius does not depend on its center.If is a code with minimum distance and is a spherewith radius then the following is readily verified.

Theorem 1: For a code with minimum distance anda sphere with radius , we have .

Theorem 1 is known as the sphere packing bound. In a codewhich attains the sphere packing bound, i.e., ,

the spheres with radius around the codewords of form apartition of . Hence, such a code is a perfect code. A perfectcode with radius is also called a perfect -error-correctingcode.Finally, a code of length which attains the sphere packing

bound is determined by three parameters, its length, the radius ofthe sphere , which is equivalent to the fact that hasminimumdistance ; the size of the code is , where is the spaceon which the code is defined and is a sphere with radius .It is clear from the sphere packing bound that it is important to

compute the size of a sphere in any given metric. We start withthe size of a sphere in the Hamming scheme. For two words

and over theHamming distance, is defined by

Lemma 1: The size of a sphere with radius centered in aword of length over in the Hamming scheme is

Corollary 1: Let be a perfect single-error-correcting codeof length over an alphabet with letters in the Hammingscheme. Then, the size of is .

Corollary 2: If a code of length over an alphabet withletters has minimum Hamming distance and sizethen is a perfect code.

An -dimensional Lee sphere with radius , centeredat is the shape in such that

if and only if , i.e.,it consists of all points in whose Manhattan distance from

ETZION: PRODUCT CONSTRUCTIONS FOR PERFECT LEE CODES 7475

the given point is at most . The size ofis well known [3]

(2)

The -dimensional Lee sphere with radius is the spherewith radius in theManhattan metric and also in the Lee metricwhenever the alphabet satisfies .

Corollary 3: Let be a perfect single-error-correcting codeof length over in the Lee metric. Then, the size of is

.

Corollary 4: If a code of length over has minimumLee distance three and size then is a perfect code.

If the code is over and the code is a perfect -error-cor-recting code then the spheres with radius around the code-words form a tiling of . Instead of Theorem 1, we have thefollowing theorem.

Theorem 2: For a code with minimum Manhattandistance , whose codewords are the points of a latticewe have .

B. Perfect Codes in the Hamming Scheme

The existence question of perfect codes in the Hammingscheme was well investigated. It is well known [33] that theonly parameters for nontrivial perfect codes over arethose of the two Golay codes and the Hamming codes. Alsoover other alphabets it is highly probable that no new perfectcodes exist [34]. The Hamming codes have length , ,where the alphabet is . The minimum Hamming distanceof the code is three. Codes with the parameters of the Hammingcodes were extensively studied. A small sample of referencesincludes [20]–[26].An code over is a linear subspace

of dimension of . Clearly, has codewords andcosets. For any code (linear or nonlinear) of length over

and a word we define a translate ofwith the word by

If is a linear code then each translate is a coset of the code. It iseasy to verify that the size of a translate is equal to the sizeof . If is linear and , then either or

. Therefore, the cosets of the code forma partition of . If is nonlinear then this is usually not true.But, if is a perfect code then some translates form a partition of. For example, if is a perfect single-error-correcting code

and and are two different unit vectors of length , then. Therefore, the code and all its

translates formed from the unit vectors form a partition of .Given a binary perfect single-error-correcting code of

length , one can define the extended code by

where is the parity of , i.e., zero if the weight of (numberof ones in ) is even and one if the weight of is odd. Theextended code has length , size exactly as the size of ,and minimum Hamming distance four. All the codewords ofhave even weight. The extended code has odd translates(all words have odd weight) consisting of all translates fromunit vectors. It has even translates (all words have evenweight) consisting of the code and all translates from vectorsof weight two with an one in the last coordinate. All thesetranslates are disjoint and together form a partition of . Abinary perfect single-error-correcting code has lengthand codewords. Let be a code of length , minimumdistance four, and codewords all with even weight. It iseasy to verify that the punctured code

has length , minimum distance three, and codewords.Hence, by Corollary 2, is a perfect code.

C. Perfect Codes in the Lee and Manhattan Metrics

The existence question of perfect -error-correcting codes inthe Lee metric was first asked by Golomb and Welch in theirseminal paper [3]. They constructed perfect codes for length

over , where , and the minimumLee distance of the code is . They also constructed perfectsingle-error-correcting codes of length over , .They also considered perfect codes in the Manhattan metric.They proved that for each there exists an integersuch that for each radius there is no tiling of the -di-mensional Euclidian space with the sphere , i.e., there isno length , perfect -error-correcting code in the Manhattanmetric. The existence problem of other perfect codes was dis-cussed in many papers, e.g., [3], [8]–[10], and [12]. We willmention two of the results given in these papers. Post [9] provedthat if then there are no perfect Lee codes over

for , , and for ,. Another interesting result was given in

[12] where it was proved that the smallest for which thereexists a perfect single-error-correcting Lee code over is themultiplication of all the prime factors of .In the same way as was done for the Hamming scheme, a

perfect single-error-correcting code (in the Lee metric) andits translates formed from the unit vectors (vectors of lengthwith exactly one nonzero entry having value or ) form apartition of . If the perfect code is over then the partitionis of . In contrary to the Hamming scheme, not many con-structions are known for perfect single-error-correcting codesin the Lee metric. Not many properties are known, and there isno ”reasonable” lower bound on the number of nonequivalentperfect single-error-correcting codes.Finally, if , then is not the sphere in the

corresponding parameters of the Leemetric. If or ,then the Lee metric coincides with the Hamming scheme forbinary codes or ternary codes, respectively. If , then thesituation becomes more interesting, but also more complicated.


D. Anticodes and Diameter Perfect Codes

In all the perfect codes the minimum distance of the code isan odd integer. If the minimum distance of the code is aneven integer then there cannot be any perfect code since forany two codewords such that thereexists a word such that and . Forthis case another concept is used, a diameter perfect code, aswas defined in [29]. This concept is based on the code–anticodebound presented by Delsarte [35]. An anticode of diameterin a space is a subset of words from such thatfor all .

Theorem 3: If a code , in a space of a distance regulargraph, has minimum distance and in the anticode of thespace the maximum distance is then .

Theorem 3 which is proved in [35] is applied to the Ham-ming scheme since the related graph is distance regular. Itcannot be applied to the Manhattan metric since the relatedgraph is not finite. It cannot be applied to the Lee metricsince the related graph is not distance regular. This can beeasily verified by considering the three words ,

, and of length over ,. ; there exists exactly one

word for which and , while thereare exactly two words of the form for whichand . Fortunately, an alternative proof which wasgiven in [29] can be slightly modified to work for the Leemetric.

Theorem 4: Let be a code of length over with Leedistances between codewords taken from a set . Letand let be the largest code in with Lee distances betweencodewords taken from the set . Then

Proof: Let .For a given codeword and a word , there isexactly one element such that .Therefore, .Since is the largest code in , with Lee distances between

codewords taken from the set , it follows that for any givenword the set has at mostcodewords. Hence, .Thus, and the claim is proved.

Corollary 5: Theorem 3 holds for the Lee metric, i.e., if acode has minimum Lee distance and in the anticode

the maximum Lee distance is , then.Proof: Let and let be a

code from with minimum Lee distance . Let be a subsetof with Lee distances between words of taken from theset . i.e., is an anticode with diameter .Clearly, the largest code in with Lee distances from hasonly one codeword. Applying Theorem 4 on , , and ,implies , i.e., .

Thus, the claim is proved.

Corollary 5 can be stated similarly to Theorem 2.

Theorem 5: Let be a lattice which forms a codewith minimumManhattan distance . Then, the size of any anti-code of length with maximum distance is at most .

A code which attains either the bounds of Theorem 3 or thebound of Corollary 5 or the bound of Theorem 5 with equality iscalled a diameter perfect code. The definition in a space whichconsists of the vertices from a distance regular graph was firstgiven by Ahlswede et al. [29].What is the size of the largest anticode with Diameter in, ? The answer was given in [36, pp. 30–41]. If

then this anticode is an -dimensional Lee sphere with radius ,. For odd we define anticodes with diameter

as follows. For , let be a shape which consists oftwo adjacent points of , where two points are adjacent if theManhattan distance between them is one. is defined byadding to all the points which are adjacent to at least one ofits points. is the largest anticode in with diameter. and were used in [17] for 2-D interleaving schemesto correct 2-D cluster errors. Similarly to the computation of (2)in [3], one can compute the size of [27] and obtain

(3)

Ahlswede et al. [29] have examined the Hamming, Johnson,and Grassmann schemes for the existence of diameter perfectcodes. All perfect codes in these schemes are also diameter per-fect codes. In the Hamming schemes, there are two more fami-lies of diameter perfect codes.MDS codes [33] are diameter per-fect codes, extended Golay codes, and extended binary perfectsingle-error-correcting codes are also diameter perfect codes.Finally, similarly to Corollaries 2 and 4 we have the following

result.

Lemma 2: If a code of length over has minimum Leedistance four and size , then is a diameter perfect code.

E. Diameter Perfect Codes in the Lee Metric

By (3), the size of a maximum anticode of length with di-ameter three over in the Manhattan metric is . Therefore,by Theorem 5, a related diameter perfect code with minimumdistance four can be formed from a lattice with volume inwhich the minimum Manhattan distance is four. Consider thelattice defined in [27] by the following generator matrix:

where , is the matrix for which, is an all-zero matrix,

and .


Example 1: For , has the form

The volume of is and it is easy to verify that theminimum Manhattan distance of the code defined by isfour. Moreover, it is readily verified that is reduced to acode over . Hence, we have the following theorem.

Theorem 6: The code defined by the lattice is a diam-eter perfect code, with minimum distance four, in theManhattanmetric. The code can be reduced to a Lee code over .is a diameter perfect code, with minimum distance four, in theLee metric.

The code defined by is not unique. There are otherdiameter perfect codes of length and minimum Lee distancefour. Some of these codes will be considered in Section IV. Themain goal will be to construct many such codes over alphabetswith the smallest possible size.For there are diameter perfect codes for each even

minimum Lee distance. By (3), the size of a maximum anticodeof length 2 with diameter over is . For each

the following generator matrix forms a latticewhich generates a diameter perfect code with minimum distance.

These lattices will be interleaved together to form more diam-eter perfect codes and nonperiodic diameter perfect codes inSection V.Cosets and translates of a code are defined in the Lee

and Manhattan metrics in the same way that they are de-fined in the Hamming scheme. The Lee (Manhattan) weightof a word over is defined by

, where is the all-zero word. A translateof a code is said to be an even translate if all its words haveeven Lee (Manhattan) weight. A translate is said to be an oddtranslate if all its words have odd Lee (Manhattan) weight.Similarly to extended perfect single-error-correcting codes in

the Hamming schemewe have even translates and odd translatesfor diameter perfect codes with even distance. This is proved inthe following results. The first lemma can be readily verified.

Lemma 3: A diameter perfect code of length , over ,with minimum Lee distance can be extended to a diameterperfect code of length , over , with minimum Manhattandistance .

Lemma 4: If is a diameter perfect code with minimumManhattan distance , even, then all the codewords of haveeven Manhattan weight.

Proof: Let be a diameter perfect code with minimumdistance , in the Manhattan metric. Assume the contrary

that there exists a codeword with odd Manhattan weight. Sinceis a perfect diameter code it follows that there exists a tiling ofwith the anticode whose diameter is . W.l.o.g.,

we can assume that the two adjacent points in (the core ofeach in this tiling) differ in the last coordinate: one of thesepoints is a codeword in and the secondpoint is . In the tiling there exists twopointssuch that for which is in an anticode whichcontains the codeword and is in an anticode whichcontains the codeword ; and furthermore, has evenManhattan weight and has odd Manhattan weight. Clearly,

, is odd and atleast . It implies thatand .and the facts that in the tiling is contained in the anticodeof and is contained in the anticode of implies thatis greater than the last entry of and is greater than thelast entry of . It follows that

and sincewe have that , a contradiction.Thus, if is a diameter perfect code with minimum Man-

hattan distance , even, then all the codewords of have evenManhattan weight.

Lemma 5: If is a diameter perfect code with minimum Leedistance , even, over , then is even.

Proof: By (3), the size of the anticode with maximum dis-tance is even. This implies by the definition of a diameterperfect code that the size of the space is even. Thus, is even.

Corollary 6: If is a diameter perfect code with minimumLee distance , even, then all the codewords of have evenLee weight.

Theorem 7: Each translate of a diameter perfect code witheven distance, in the Lee metric, is either an even translate or anodd translate. The number of even translates is equal the numberof odd translates.

Proof: Let be a diameter perfect code in the Lee metric.By Lemma 5 and Corollary 6 each translate of is either eventranslate or an odd translate. W.l.o.g., we can assume that thetwo points of in are and

. Let be a diameter perfect code of length anddiameter , in the Lee metric, where . Since issymmetric around and we have that for each ,

, and for each point , .Therefore, the number of even translates of is equal to thenumber of odd translates of .

We note one important difference between binary perfectcodes and binary diameter perfect codes in the Hammingscheme and perfect codes and diameter perfect codes in theLee metric. While binary perfect single-error-correcting codesin the Hamming scheme can be extended to binary diameterperfect codes (and vice versa via puncturing) by adding a paritybit (removing a coordinate) and thus increasing (decreasing)the distance by one, this cannot be done in the Lee metric.


Finally, there is one more diameter perfect code in the Leeand Manhattan metric. It is formed by the Minkowski lattice[37] with the generator matrix

The related anticode in this case is .

F. A Doubling Construction for Binary Perfect Codes

Let be a binary perfect single-error-correcting code oflength and let be its extended code of length. Let be its translates and let

be the even translates of its extendedcode. Let be the code constructed as follows:

The following was proved by Phelps [20].

Theorem 8: is a perfect single-error-correcting code oflength .

It is readily verified that the perfect extended code is thecode defined by

Let be a binary perfect single-error-correcting code oflength and let be its extended code of length. Let be the even translates of

its extended code. Let bea permutation of . The following theorem is animmediate consequence

Theorem 9: The code defined by

is an extended perfect code of length with minimum Ham-ming distance four.

Note that the first element in the permutation is to makesure that the all-zero word will be a codeword in .

III. A NEW PRODUCT CONSTRUCTION FOR -ARYPERFECT CODES

There are several product constructions for nonbinary per-fect codes in the Hamming scheme. Most notable is the generalconstruction of Phelps [22]. Another construction was given byMollard [23]. The construction that we present is a generaliza-tion for a construction of Zinov’ev [24].We will present a new simple construction, for perfect codes

in the Hamming scheme, which will be very effective in con-structions of perfect codes in the Lee metric. For our construc-tion, we will use two perfect codes in the Hamming scheme. Thefirst code is a perfect single-error-correcting code of length

over an alphabet with letters, which have a total of

translates, including itself. The second code is a per-fect single-error-correcting code of length over analphabet with letters. Let be the th translateof , where . We construct the following code :

Theorem 10: The code is a -ary perfect single-error-cor-recting code of length .

Proof: Clearly, the length of the codewords fromis . We will first prove that the min-imum distance of is three. Let and

be two distinct codewords of . We nowdistinguish between two cases.Case 1) If , then

, sinceand

. W.l.o.g., we can assume that ,, and , and hence ,, and , which implies that

.Case 2) If , then there exists

a , such that and since(i.e., ), , and, it follows that and therefore

.Thus, .By Corollary 1, the size of is and the

size of is . Clearly,

.This implies by Corollary 2 that is a -ary perfect single-

error-correcting code of length .

IV. CONSTRUCTIONS FOR PERFECT AND DIAMETERPERFECT LEE CODES

In this section, we will modify the two product constructionsfor perfect codes and diameter perfect codes in the Hammingscheme to obtain perfect codes and diameter perfect codes inthe Lee and Manhattan metrics.

A. Diameter Perfect Codes With Minimum Distance Four

Let and be diameter perfect codes. Eachcode has translates from which are even translates. Let

, , be these even translates.Let be a permutation of

. The following theorem is a modification of The-orem 9.

Theorem 11: The code defined by

is a diameter perfect Lee code of length , over , with min-imum Lee distance four.

Proof: The size of the code , , is and hencethe size of is . The Lee distance of the codeis four as the one given in Theorem 9. Since no proof is given


in Section II-F we will now present a proof. Let andbe two distinct codewords in such that and. We distinguish between two cases.

Case 1) If , then and . Hence,, , which implies that

.Case 2) If , then and .

Since or , we have thator , respectively.

Hence, .Thus, .The code is defined over and hence the size of its

space is . Since by (3) the size of the related anticode withdiameter four is and the minimum Lee distance of is four,it follows by Lemma 2 that is a diameter perfect Lee code oflength , over , with minimum Lee distance four.

Corollary 7: Let , where is an odd prime greaterthan one. There exists an diameter perfect code.The lattice of the following generator matrix:

forms a diameter perfect code. By applying Theorem11 iteratively we obtain the following theorem.

Theorem 12: For each , , there exists andiameter perfect code.

Codes with the same parameters as in Theorem 12 were gen-erated by Krotov [38].Theorem 11 can be modified and applied on codes in with

the Manhattan metric.

Theorem 13: Let and be diameter perfect codesof length with minimum Manhattan distance four. Eachcode has translates from which are even translates. Let

, , be these even translates.Let be a permutation of

. The code defined by

(4)

is a diameter perfect Manhattan code of length , over , withminimum Manhattan distance four.

B. Perfect Single-Error-Correcting Lee Codes

In this subsection, we use a modification of the productconstruction of Section III to construct perfect single-error-cor-recting Lee codes.Let be a perfect single-error-correcting Lee code of length

, odd, over an alphabet with letters, whichhas a total of translates (the size of the Lee sphere), includingitself. Let

, be a permutation of . We have permutation,where the first permutation is the identity permutation. Let bea perfect single-error-correcting Hamming code of length

over an alphabet with letters. Let ,be the th translate of , where . We construct thefollowing code :

Theorem 14: The code is a perfect single-error-correctingLee code of length over an alphabet of size .

Proof: Clearly, the length of the codewords from is. The proof that the code has minimum

Lee distance three is identical to the related proof in Theorem10.By Corollary 1, the size of is .

The size of is . Clearly,

.This implies by Corollary 4 that is a perfect single-error-

correcting Lee code of length over an alphabet of size.

V. ON THE NUMBER OF PERFECT CODES AND NONPERIODICCODES IN THE MANHATTAN METRIC

In this section, we consider two problems related to perfectcodes. The first one is the number of nonequivalent perfectcodes, which was considered in many papers for the Hammingschemes, e.g., [22] and [26]. It was proved that the number ofnonequivalent perfect single-error-correcting code of lengthover is , where is a constant, . The

second problem is whether there exist nonperiodic perfectand diameter perfect codes in with the Manhattan metric.Periodic and nonperiodic codes were mentioned in [12], but forthe best of our knowledge no construction of nonperiodic codeswas known until recently. We note that we ask the question onnonperiodic codes in the Manhattan metric.We will not go into the more complicated exact computations

on the number of nonequivalent perfect and diameter perfectcodes. We will just count the number of different perfect codes.The two product constructions given in Section IV can be usedto provide a lower bound on the number of perfect codes anddiameter perfect codes in the Lee metric. Given a diameter per-fect code of length , prime, the size of the related anticode is, and hence there are even translates. Therefore, there are

different perfect codes of length , given in the con-struction of Theorem 11. Continuing iteratively with the con-struction of Theorem 11, we obtain

different diameter perfect codes of length over .Similarly, a bound on the number of perfect Lee code can be

obtained from the construction of Section IV-B. In this compu-tation, we can also take into account the bounds on the numberof perfect codes in the Hamming scheme which are used in theconstruction.Before we continue to discuss the topic of nonperiodic perfect

codes and diameter perfect codes in we consider a questionrelated to both problems discussed in this section.Let and be two distinct subcodes of , such that any

elements in is within Manhattan distance one from a uniquecodeword of and a unique codeword of . If we consideranticodes instead of spheres then the elements of and arecenters of the anticodes which form a tiling of . If is con-tained in a perfect single-error-correcting Manhattan code (or adiameter perfect code with minimum Manhattan distance four)


then it can be replaced by to obtain a new different per-fect single-error-correcting Manhattan code (or a diameter per-fect code with minimumManhattan distance four) . The tech-nique was used in the Hamming scheme to form a large numberof nonequivalent perfect codes [25], [26]. The same techniquecan be considered for the Lee metric.The constructions in Section IV can provide such a space

and codes and . If we change the permutation of the lastcoordinate by one transposition, it is easily verified that the in-tersection of the two generated codes is relatively large (usuallyabout of the code size, where for the construc-tion of the diameter perfect codes and for the construc-tion of the perfect single-error-correcting code). The two partswhich are not in the intersection will have the role of and. These can be the building block for nonperiodic codes in

the Manhattan metric. This is left as a problem for future re-search. Unfortunately, such a finite space and codesand cannot exist. A subset is perfectly -covered by iffor each element there is a unique element suchthat .

Theorem 15: There is no finite subset of that can be per-fectly -covered by two different codes, using the Manhattanmetric.

Proof: Assume the contrary; let be a subset of ofsmallest possible size, such that there exist two codes andwhich -cover . Let be the largest integer for which

there exists a codeword in or with the value in one ofthe coordinates. W.l.o.g., we can assume that such a code-word is . This codeword covers the word

. Since can be the largest integerin a codeword of , then the only codeword of which cancover is . Thus,

and perfectly -covera subset , a contradiction to the assumption that is suchsubset with the smallest size. Thus, there is no finite subset inthat can be perfectly -covered by two different codes.

For each , , there exists a nonperiodic diameterperfect code. If , then such a code exists for each evenminimum Manhattan distance , . We are usinginterleaving of the lattices used to construct diameterperfect codes with minimum Manhattan distance . For agiven , let be an infinite sequence, where

. Given , we construct the following set of points:

The sequence will be called nonperiodic if thereare no nonzero integer and an integer such that

.

Theorem 16: If the points of the set are taken as centers forthe spheres , then we obtain a tiling. The tiling is nonperi-odic if the sequence is nonperiodic.

Proof: First, we note that the set of centersforms a connected nonintersecting diagonal strip

with the spheres . The same is true for the set of centersfor any given

. The set of centers

is exactly the lattice formed from the generatormatrix

which forms a diameter perfect code with minimum distance. Finally, replacing the set of centers

in this tiling by the set of centersis just a

shift of the diagonal strip in a 45 diagonal direction. This doesnot affect the fact that the set of these centers forms a tiling withthe spheres .It is readily verified that if the sequence is nonperiodic, then

also the tiling generated from is nonperiodic.

The construction of 2-D nonperiodic diameter perfect codesyields an uncountable number of diameter perfect codes. Thenumber of nonequivalent perfect codes formed in this way isequal the number of infinite sequences over the alphabet .Finally, it is easy to verify the following theorem.

Theorem 17: Let be a nonperiodic diameter perfect codeof length with minimum Manhattan distance four, and letbe a diameter perfect code of length and minimumManhattandistance four. The code defined in (4) is a nonperiodic diam-eter perfect code of length and minimumManhattan distancefour.

VI. CONCLUSION AND OPEN PROBLEMS

We have considered several questions regarding perfect codesin the Lee and Manhattan metrics. We gave two product con-structions for perfect and diameter perfect codes in these met-rics. One product construction is based on a new product con-struction for -ary perfect codes in the Hamming scheme. Thesecond product construction is based on the well-known dou-bling construction for binary perfect codes in the Hammingscheme. We used our constructions to find a lower bound onthe number of perfect and diameter perfect codes in the Lee andManhattan metrics. We also used the constructions to prove theexistence of nonperiodic perfect diameter codes in . Our dis-cussion raises many open problem fromwhich we choose a non-representative set of problems.1) Find new product constructions for perfect Lee codes anddiameter perfect Lee codes with different parameters fromthose given in our discussion.

2) Prove that there are no nontrivial perfect codes, of length, with radius greater than one in the Manhattan

metric.3) Do there exist more diameter perfect codes with minimumdistance greater than four in the Lee metric? Minkowski’slattice is the only known example.

4) For each , what is the smallest for which thereexists diameter perfect codes?

5) A tiling of is called regular if neighboring anticodesmeet along entire -dimensional faces of the originalcubes. Nonregular tiling of exists if and only ifis not a prime [39]. Does there exist a nonregular tiling offormed from anticodes with diameter three?


6) Are there values of for which there are no two nonequiv-alent perfect single-error-correcting Lee codes of length ?

7) Does there exist a nonperiodic diameter perfect code withminimum Manhattan distance four for each length greaterthan one? Recently, such a construction was given in [40]for nonperiodic perfect single-error-correcting codes oflength , where is not a prime, in the Manhattanmetric.

ACKNOWLEDGMENT

The author would like to devote this work for the memoryof the late Rudolph Ahlswede with whom he had several inter-esting discussions during the last 15 years on anticodes and per-fect codes. During an ITW meeting in Ireland at the beginningof September 2010 he brought to his attention that the size of amaximum anticode in the Manhattan metric was found recently.This made it possible to call the codes in the Lee metric obtainedin this paper, diameter perfect codes. The author would also liketo thank P. Horak and D. Krotov for their important commentsand for providing him some important references.

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Tuvi Etzion (M’89–SM’94–F’04) was born in Tel Aviv, Israel, in 1956. He re-ceived the B.A., M.Sc., and D.Sc. degrees in computer science from the Tech-nion—Israel Institute of Technology, Haifa, Israel, in 1980, 1982, and 1984,respectively.Since 1984 he held a position in the Department of Computer Science,

Technion, where he currently has a Professor position. During 1986–1987,he was a Visiting Research Professor with the Department of ElectricalEngineering—Systems, University of Southern California, Los Angeles.During summers 1990 and 1991, he was visiting Bellcore in Morristown,NJ. During 1994–1996, he was a Visiting Research Fellow in the ComputerScience Department, Royal Holloway College, Egham, U.K. He also hadseveral visits to the Coordinated Science Laboratory, University of Illinoisin Urbana-Champaign, Urbana, during 1995–1998, two visits to HP Bristolduring summers 1996 and 2000, a few visits to the Department of ElectricalEngineering, University of California at San Diego during 2000–2010, andseveral visits to the Mathematics Department, Royal Holloway College, during2007–2009. His research interests include applications of discrete mathematicsto problems in computer science and information theory, coding theory, andcombinatorial designs.Dr. Etzion was an Associate Editor for Coding Theory for the IEEE

TRANSACTIONS ON INFORMATION THEORY from 2006 until 2009.

product constructions for perfect lee codes

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