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S ,m m , . . . m, m, n denoted , v 2 omatln of tiling a form n nue a induces etelatcmo multi- common least the be ihdson oisof copies disjoint with lcdo h atc points lattice the on placed hc sgvnby given is which , v . . . v . . . v . . . 0 : ( . . m n v . ) C 1 h code The . n n ≤ stevlm fthe of volume the is m , a erdcdt a to reduced be can ( × stelnt fthe of length the is ξ nn ( 2 1 x V . . . i x n n 2 1 n atc tiling lattice (Λ) m , . . . , ) 1 < x , . . . , h minimum The . x ,      S matrix 1 sinversely is , 2 , Z oe l the all cover R x , . . . , m 1 Λ C n i generator n ≤ . namely, , − ihthe with ) a also has 1 i x , ∈ n ≤ G fa of S ) i ess, Z n + ∈ is is st n } 1 . 2 distance. The literature on codes in the Lee metric is very discuss two problems related to perfect codes and diameter extensive, e.g. [4], [9], [10], [11], [12], [13], [14], [15], perfect codes in the Lee and Manhattan metrics. The first [16]. The interest in Lee codes has been increased in the one is the number of nonequivalent such codes and the last decade due to many new and diverse applications of second one is the existence of non-periodic perfect codes these codes. Some examples are constrained and and partial- and non-periodic diameter perfect codes. We will show how response channels [4], interleaving schemes [17], multidi- our constructions can be used in the context of these two mensional burst-error-correction [18], and error-correction problems. A summary and a list of questions for future for flash memories [19]. The increased interest is also due research are given in Section VI. to new attempts to settle the existence question of perfect codes in these metrics [8]. II. BASIC CONCEPTS AND CONSTRUCTIONS Perfect codes is one of the most fascinating topics in A. Spheres and perfect codes coding theory. A perfect code in a given metric is a code in which the set of spheres with a given radius R around The main two concepts in this paper are perfect codes its codewords form a partition of the space. These codes and codes in the Lee and the Manhattan metrics. We start were mainly considered for the Hamming scheme, e.g. [20], with a general definition of perfect codes. For a given space [21], [22], [23], [24], [25], [26]. They were also considered , with a distance measure d, a subset C of is a perfect V V for other schemes as the Johnson scheme and the Grassmann code with radius R if for every element x there exists a ∈V scheme, But, as said, these codes were considered to a larger unique codeword c C such that d(x, c) R. For a point ∈ ≤ extent also in the Lee and the Manhattan metrics. x , the sphere of radius R around x, S(x, R), is the ∈ V This paper was motivated by some basic concepts which set of elements in such that y S(x, R) if and only if V ∈ were presented in [27], [28]. There are two goals for this d(x, y) R, i.e. S(x, R) = y : d(x, y) R . For ≤ { ∈ V ≤ } research. The first one is to present product constructions the sphere S(x, R), x is called the center of the sphere. In for perfect codes in the Lee and Manhattan metrics in a this paper we consider only the Hamming scheme, the Lee similar way to what was done in the Hamming scheme. The and the Manhattan metrics, in which the size of a sphere second one is to show how these constructions can be used does not depend on the center of the sphere. Hence, in the to solve other problems related to perfect codes in the Lee sequel we will assume that in our metric the size of a sphere and the Manhattan metrics. One problem is the number of with radius R does not depend on its center. If C is a code different perfect codes and diameter perfect codes in the Lee with minimum distance 2R+1 and S is a sphere with radius and Manhattan metrics. A second problem is the existence R then it is readily verified that of non-periodic perfect codes in the Manhattan metric. In Theorem 1: For a code C with minimum distance 2R+1 the process, we will also present a new product construction and a sphere S with radius R we have C S . | | · | | ≤ |V| for perfect codes in the Hamming scheme. Theorem 1 known as the sphere packing bound. In a code The rest of this paper is organized as follows. In Section II C which attains the sphere packing bound, i.e. C S = , we introduce the basic concepts which are used in our paper. the spheres with radius R around the codewords| |·| of |C form|V| We first define a perfect code in general and discuss some a partition of . Hence, such a code is a perfect code. A of the known results on perfect codes and extended perfect perfect code withV Radius R is also called a perfect R-error- codes in the Hamming scheme and the known results on correcting code. perfect codes in the Lee and Manhattan metrics. We con- Finally, a code C of length n which attains the sphere tinue by introducing the concept of anticodes and diameter packing bound is determined by three parameters, its length, perfect code as was first discussed in [29]. We prove that the radius of the sphere R which is equivalent to the fact the definition of diameter perfect codes can be applied to that C has minimum distance 2R +1; the size of the code codes in the Lee and Manhattan metrics. We discuss the is |V|S , where is the space on which the code is defined knowledge on these codes. We present a simple construction and| S| is a sphereV with radius R. for diameter perfect codes with minimum distance four It is clear from the sphere packing bound that it is impor- in the Lee and Manhattan metrics. Finally, we describe a tant to compute the size of a sphere in any given metric. We doubling construction for perfect codes in the Hamming start with the size of a sphere in the Hamming scheme. For scheme which was given by Phelps in [20]. In Section III two words x = (x1, x2,...,xn) and y = (y1,y2,...,yn) we present a new product construction for q-ary perfect over Zm the Hamming distance, dH (x, y) is defined by codes in the Hamming scheme. Similar idea to the one in this construction was used in other papers, e.g. [24], [30], dH (x, y)= i : xi = yi, 1 i n . [31], [32], for construction of relatively large binary codes |{ 6 ≤ ≤ }| (perfect and non-perfect) with minimum Hamming distance Lemma 1: The size of a sphere with radius R centered in three in the Hamming scheme. In Section IV we present a word of length n over Zm in the Hamming scheme is two product constructions, one for perfect codes and one for R n diameter perfect codes in the Lee and Manhattan metrics. (m 1)i . These constructions are modifications of the two product i  − Xi=0 constructions in the Hamming scheme. In Section V we 3

Fn Corollary 1: Let be a perfect single-error-correcting Therefore, the cosets of the code form a partition of q . code of length n overC an alphabet with m letters in the If is nonlinear then this is usually not true. But, if is n m C CFn Hamming scheme. Then the size of is 1+(m 1)n . a perfect code then some translates form a partition of q . Corollary 2: If a code of lengthC n over− an alphabet For example, if is a perfect single-error-correcting code with m letters has minimumC Hamming distance 3 and size and e and e areC two different unit vectors of length n then n i j m ∅ 1+(m 1)n then is a perfect code. (ei + ) (ej + )= . Therefore, the code and all its − C translatesC ∩ formedC from the unit vectors form aC partition of An n-dimensional Lee sphere with radius R, centered Fn n q . at (γ1,γ2,...,γn) is the shape Sn,R in Z such that n Given a binary perfect single-error-correcting code of (x1, x2, ..., xn) Sn,R if and only if Σi=1 xi γi R, C ∈ | − | ≤ length n, one can define the extended code e by i.e., it consists of all points in Zn whose Manhattan distance C n n from the given point (γ1,γ2, ..., γn) Z is at most R. The def ∈ e = (c,p(c)) : c , p(c)= c1 (mod 2) , size of Sn,R is well known [3]: C { ∈C } Xi=1 min n,R { } i n R where p(c) is the parity of c, i.e., zero if the weight of c Sn,R = 2 (2) | |  i i  (number of ones in c) is even and one if the weight of c is Xi=0 odd. The extended code e has length n +1, size exactly The n-dimensional Lee sphere with radius R is the sphere as the size of , and minimumC Hamming distance four. All with radius R in the Manhattan metric and also in the Lee the codewordsC of have even weight. The extended code Z e metric whenever the alphabet m satisfies m 2R +1. has n +1 oddC translates (all words have odd weight) C ≥ e Corollary 3: Let be a perfect single-error-correcting Cconsisting of all translates from unit vectors. It has n +1 code of length n over Zm in the Lee metric. Then the size n even translates (all words have even weight) consisting of the of C is m . 1+2n code e and all translates from vectors of weight two with Corollary 4: If a code C of length n over Zm has C mn an one in the last coordinate. All these 2n +2 translates minimum Lee distance 3 and size then C is a perfect Fn+1 1+2n are disjoint and together form a partition of 2 . A binary code. perfect single-error-correcting code has length 2r 1 and n r C r − If the code C is over Z and the code C is a perfect R- 2 − codewords. Let ′ be a code of length 2 , minimum n Cr error-correcting code then the spheres with radius R around distance four, and 2 − codewords all with even weight. It the codewords form a tiling of Zn. Instead of Theorem 1 is easy to verify that the punctured code we have the following theorem. pdef = c : (c,p(c)) ′ Theorem 2: For a code C Zn with minimum Manhat- C { ∈C } ⊆ r n r tan distance 2R +1, whose codewords are the points of a has length 2 1, minimum distance three, and 2 − − p lattice Λ we have V (Λ) Sn,R . codewords. Hence, by Corollary 2, is a perfect code. ≥ | | C B. Perfect codes in the Hamming scheme C. Perfect codes in the Lee and Manhattan metrics The existence question of perfect codes in the Hamming The existence question of perfect R-error-correcting codes scheme was well investigated. It is well known [33] that the in the Lee metric was first asked by Golomb and Welch only parameters for nontrivial perfect codes over GF(q) are in their seminal paper [3]. They constructed perfect codes 2 those of the two Golay codes and the Hamming codes. Also for length n = 2 over Zm, where m = 2R +2R + 1, over other alphabets it is highly probable that no new perfecr t and the minimum Lee distance of the code is 2R + 1. q 1 codes exist [34]. The Hamming codes have length q −1 , They also constructed perfect single-error-correcting codes r 2, where the alphabet is GF(q). The minimum Hamming− of length n over Z , m n . They also considered ≥ m = 2 +1 distance of the code is three. Codes with the parameters perfect codes in the Manhattan metric. They proved that of the Hamming codes were extensively studied. A small for each n > 3 there a exists an integer ρn such that for sample of references includes [20], [21], [22], [24], [23], each radius R>ρn there is no tiling of the n-dimensional [25], [26]. Euclidian space with the sphere Sn,R, i.e., there is no length F An [n,k,d] code over q =GF(q) is a linear subspace n, perfect R-error-correcting code in the Manhattan metric. FnC k n k of dimension k of q . Clearly, has q codewords and q − The existence problem of other perfect codes was discussed C cosets. For any code (linear or nonlinear) of length n over in many papers, e.g. [3], [8], [9], [10], [12]. We will mention C n Fq =GF(q) and a word v Fq we define a translate of two of the results given in these papers. Post [9] proved ∈ C with the word v by that if m 2R +1 then there are no perfect Lee codes ≥ def over Z for 3 n 5, R n 2, and for n 6, v + = v + c : c . m √2 1 ≤√ ≤ ≥ − ≥ C { ∈ C} R 2 n 4 (3 2 2). Another interesting result was If is a linear code then each translate is a coset of the given≥ in [12]− where it− was proved that the smallest m for code.C It is easy to verify that the size of a translate v + which there exists a perfect single-error-correcting Lee code FCn Z is equal to the size of . If is linear and v1, v2 q over m is the multiplication of all the prime factors of then either v + = v C+ orC (v + ) (v + )∈ = ∅. 2n +1. 1 C 2 C 1 C ∩ 2 C 4

Zn In the same way as was done for the Hamming scheme, exactly one element v = c + α m such that α = c + v. a perfect single-error-correcting code C (in the Lee metric) Therefore, S = C . ∈ | | | D| · |A| and its translates formed from the unit vectors (vectors of Since C′ is the largest code in , with Lee distances length n with exactly one nonzero entry having value -1 or between codewordsD taken from the setA , it follows that for Zn Z Zn D C +1) form a partition of m. If the perfect code is over then any given word v m the set c : c , c + v n ∈ { ∈ D n ∈ A} the partition is of Z . In contrary to the Hamming scheme, has at most C′ codewords. Hence, S C′ m . | D| n | | ≤ | D| · not many constructions are known for perfect single-error- Thus, C C′ m and the claim is proved. | D| · |A| ≤ | D| · correcting codes in the Lee metric. Not many properties Corollary 5: Theorem 3 holds for the Lee metric, i.e. if are known, and there is no ”reasonable” lower bound on C Zn a code m, has minimum Lee distance d and in the the number of nonequivalent perfect single-error-correcting ⊂ Zn anticode m the maximum Lee distance is d 1 then codes. C A⊂mn. − | | · |A| ≤ Finally, if m < 2R +1 then Sn,R is not the sphere in m C Proof: Let = d, d +1,..., 2 n and let be the corresponding parameters of the Lee metric. If m = 2 ZnD { ⌊ ⌋ } D a code from m with minimum Lee distance d. Let be or m =3 then the Lee metric coincides with the Hamming Zn A a subset of m with Lee distances between words of scheme for binary codes or ternary codes, respectively. If taken from the set 1, 2,...,d 1 . i.e. is an anticodeA m> 3 then the situation becomes more interesting, but also with diameter d {1. Clearly, the− } largestA code in with more complicated. Lee distances from− has only one codeword. ApplyingA D CD 1 Theorem 4 on , C , and , implies | n| , i.e. D D A m ≤ C mn. |A| D. Anticodes and diameter perfect codes | D| · |A| ≤ Thus, the claim is proved. In all the perfect codes the minimum distance of the Corollary 5 can be stated similarly to Theorem 2. code is an odd integer. If the minimum distance of the Theorem 5: Let Λ be a lattice which forms a code C code C is an even integer then there cannot be any perfect Zn with minimum Manhattan distance d. Then the size of⊂ code since for any two codewords c ,c C such that 1 2 any anticode of length n with maximum distance d 1 is d(c ,c )=2δ there exists a word x such that∈ d(x, c )= δ 1 2 1 at most V (Λ). − and d(x, c2) = δ. For this case another concept is used, a diameter perfect code, as was defined in [29]. This A code which attains either the bounds of Theorem 3 or concept is based on the code-anticode bound presented by the bound of Corollary 5 or the bound of Theorem 5 with Delsarte [35]. An anticode of diameter D in a space equality is called a diameter perfect code. The definition is a subset of words from Asuch that d(x, y) D for allV in a space which consists of the vertices from a distance x, y . V ≤ regular graph was first given by Ahlswede, Aydinian, and Theorem∈ A 3: If a code C, in a space of a distance regular Khachatrian [29]. graph, has minimum distance d and inV the anticode of the What is the size of the largest anticode with Diameter D space the maximum distance is d 1 then C A . in Zn, n 2? The answer was given in [36, pp. 30–41]. If V − | | · |A| ≤ |V| D =2R ≥then this anticode is an n-dimensional Lee sphere Theorem 3 which is proved in [35] is applied to the with radius R, S . For odd D we define anticodes with Hamming scheme since the related graph is distance regular. n,R diameter D = 2R +1 as follows. For R = 0, let S be It cannot be applied to the Manhattan metric since the n,′ 0 a shape which consists of two adjacent points of Zn, where related graph is not finite. It cannot be applied to the two points are adjacent if the Manhattan distance between Lee metric since the related graph is not distance regular. them is one. S is defined by adding to S all the This can be easily verified by considering the three words n,R′ +1 n,R′ points which are adjacent to at least one of its points. S x = (000 ... 0), y = (200 ... 0), and z = (110 ... 0) of n,R′ is the largest anticode in Zn with diameter 2R +1. S length n over Z , m 5. d (x, y) = d (x, z) = 2; 2,R m L L and S were used in [17] for two-dimensional interleaving there exists exactly one word≥ u for which d (x, u)=1 and 2′ ,R L schemes to correct two-dimensional cluster errors. Similarly d (u,y)=1, while there are exactly two words of the form L to the computation of (2) in [3] one can compute the size u for which d (x, u)=1 and d (u,z)=1. Fortunately, L L of S [27] and obtain an alternative proof which was given in [29] can be slightly n,R′ modified to work for the Lee metric. C Z min n 1,R Theorem 4: Let be a code of length n over m with { − } D i+1 n 1 R +1 Lee distances between codewords taken from a set . Let S′ = 2 − . (3) | n,R|  i  i +1  Zn C D i m and let ′ be the largest code in with Lee X=0 distancesA ⊂ between codewordsD taken from the setA . Then D Ahlswede, Aydinian, and Khachatrian [29] have exam- C C′ ined the Hamming, Johnson and Grassmann schemes for | D| | D| . mn ≤ the existence of diameter perfect codes. All perfect codes |A| in these schemes are also diameter perfect codes. In the def C Zn Proof: Let S = (c, v): c , v m, c+v . Hamming schemes there are two more families of diameter For a given codeword{ c C ∈andD a word∈ α , there∈ A} is perfect codes. MDS codes [33] are diameter perfect codes, ∈ D ∈ A 5 extended Golay codes, and extended binary perfect single- Cosets and translates of a code C are defined in the Lee error-correcting codes are also diameter perfect codes. and Manhattan metrics in the same way that they are defined Finally, similarly to Corollary 2 and Corollary 4 we have in the Hamming scheme. The Lee (Manhattan) weight of the following result. a word x = (x1, x2,...,xn) over Zm (Z) is defined Lemma 2: If a code C of length n over Zm has minimum by dL(x, 0) (dM (x, 0)), where 0 is the all-zero word. A mn C C Lee distance 4 and size 4n then is a diameter perfect translate of a code is said to be an even translate if all code. its words have even Lee (Manhattan) weight. A translate is said to be an odd translate if all its words have odd Lee E. Diameter perfect codes in the Lee metric (Manhattan) weight. By (3) the size of a maximum anticode of length n with Similarly to to extended perfect single-error-correcting diameter three over Zn in the Manhattan metric is 4n. codes in the Hamming scheme we have even translates and Therefore, by Theorem 5, a related diameter perfect code odd translates for diameter perfect codes with even distance. with minimum distance four can be formed from a lattice This is proved in the following results. The first lemma can with volume 4n in which the minimum Manhattan distance be readily verified. C is four. Consider the lattice Λ(G ) defined in [27] by the Lemma 3: A diameter perfect code of length n, over n Z following generator matrix m, with minimum Lee distance d can be extended to a diameter perfect code C′ of length n, over Z, with minimum An Bn Manhattan distance d. Gn = ,  Cn Dn  Lemma 4: If C is a diameter perfect code with minimum Manhattan distance d, d even, then all the codewords of C where An = In 1, Bn is the (n 1) 1 matrix for which T − − × have even Manhattan weight. B = [3 5 2n 1], Cn is an 1 (n 1) all-zero matrix, n ··· − × − Proof: Let C be a diameter perfect code with minimum and Dn = [4n]. distance 2R + 2, in the Manhattan metric. Assume the Example 1: For n =6, G6 has the form contrary that there exists a codeword with odd Manhattan 10000 3 weight. Since C is a perfect diameter code it follows that  01000 5  n there exists a tiling of R with the anticode S′ whose 00100 7 n,R G =   diameter is 2R +1. W.l.o.g. we can assume that the two 6  00010 9    adjacent points in Sn,′ 0 (the core of each Sn,R′ in this  0 0 0 0 1 11    tiling) differ in the last coordinate, one of these points  0 0 0 0 0 24  (z1,...,zn 1,zn) is a codeword in C and the second point   − The volume of Λ(Gn) is 4n and it is easy to verify that is (z1,...,zn 1,zn +1). In the tiling there exists two points − n the minimum Manhattan distance of the code defined by x = (x1,...xn 1, xn),y = (y1,...yn 1,yn) Z such − − ∈ Λ(Gn) is four. Moreover, it is readily verified that Λ(Gn) is that dM (x, y)=1 for which x is in an anticode which reduced to a code over Z . Hence, we have the following contains the codeword α C and y is in an anticode which 4n ∈ theorem. contains the codeword β C; and furthermore, α has even ∈ Theorem 6: The code defined by the lattice Λ(Gn) is a Manhattan weight and β has odd Manhattan weight. Clearly, diameter perfect code, with minimum distance four, in the dM (x, α), dM (y,β) R, R +1 , dM (α, β) is odd and at ∈ { } Manhattan metric. The code can be reduced to a Lee code least 2R +2. It implies that dM (x, α)= dM (y,β)= R +1 C over Z4n. C is a diameter perfect code, with minimum and dM (α, β)=2R+3. dM (x, α)= dM (y,β)= R+1 and distance four, in the Lee metric. the facts that in the tiling x is contained in the anticode of α and y is contained in the anticode of β implies that x is The code defined by Λ(G ) is not unique. There are n n greater than the last entry of α and y is greater than the last other diameter perfect codes of length n and minimum Lee n entry of β. It follows that d ((x ,...,x , x 1), α)= distance four. Some of these codes will be considered in M 1 n 1 n d ((y ,...,y ,y 1),β)= R and since− d −(x, y)=1 Section IV. The main goal will be to construct many such M 1 n 1 n M we have that d −(α, β−) 2R +1, a contradiction. codes over alphabets with the smallest possible size. M Thus, if C is a diameter≤ perfect code with minimum For n = 2 there are diameter perfect codes for each Manhattan distance d, d even, then all the codewords of even minimum Lee distance. By (3) the size of a maximum C have even Manhattan weight. anticode of length 2 with diameter 2R + 1 over Z is Lemma 5: If C is a diameter perfect code with minimum 2(R + 1)2. For each i, 0 i R the following generator Lee distance d, d even, over Z , then m is even. matrix forms a lattice which≤ ≤ generates a diameter perfect m Proof: By (3) the size of the anticode with maximum code with minimum distance 2R. distance d 1 is even. This implies by the definition of R +1+ i R +1 i − − . a diameter perfect code that the size of the space is even.  i 2(R + 1) i  − Thus, m is even. These lattices will be interleaved together to form more Corollary 6: If C is a diameter perfect code with mini- diameter perfect codes and non-periodic diameter perfect mum Lee distance d, d even, then all the codewords of C codes in Section V. have even Lee weight. 6

Theorem 7: Each translate of a diameter perfect code Theorem 9: The code defined by with even distance, in the Lee metric, is either an even def i π(i) r translate or an odd translate. The number of even translates ∗ = (x, y): x , y , 1 i 2 C { ∈Be ∈Ce ≤ ≤ } is equal the number of odd translates. r+1 Proof: Let C be a diameter perfect code in the Lee is an extended perfect code of length 2 with minimum metric. By Lemma 5 and Corollary 6 each translate of Hamming distance four. C is either even translate or an odd translate. W.l.o.g. Note, that the first element in the permutation π is 1 to we can assume that the two points of Sn,′ 0 in Sn,R′ are make sure that the all-zero word will be a codeword in ∗. α = (0,..., 0, 0) and β = (0,..., 0, 1). Let C a diameter C perfect code of length n and diameter 2R +1, in the Lee C III. ANEW PRODUCTCONSTRUCTIONFOR q-ARY metric, where α . Since Sn,R′ is symmetric around α and β we have that∈ for each 0 δ R +1, x : x PERFECT CODES ≤ ≤ |{ ∈ Sn,R′ , dL(x, α)= δ = x : x Sn,R′ , dL(x, β)= δ , There are several product constructions for non-binary }| |{ ∈ }| and for each point x, dL(x, α) = dL(x, β)+1 (mod 2). perfect codes in the Hamming scheme. Most notable is the Therefore, the number of even translates of C is equal the general construction of Phelps [22]. Another construction number of odd translates of C. was given by Mollard [23]. The construction that we present We note on one important difference between binary is a generalization for a construction of Zinov’ev [24]. perfect codes and binary diameter perfect codes in the We will present a new simple construction, for perfect Hamming scheme and perfect codes and diameter perfect codes in the Hamming scheme, which will be very effective codes in the Lee metric. While binary perfect single-error- in constructions of perfect codes in the Lee metric. For our correcting codes in the Hamming scheme can be extended to construction we will use two perfect codes in the Hamming binary diameter perfect codes (and vice versa via puncturing) scheme. The first code r 1 is a perfect single-error-correcting by adding a parity bit (removing a coordinate) and thus q C 1 code of length n = q −1 over an alphabet with q letters, increasing (decreasing) the distance by one, this cannot be −r which have a total of q translates, including 1 itself. The done in the Lee metric. C second code rs2 is a perfect single-error-correcting code of Finally, there is one more diameter perfect code in the q C 1 r i length ℓ = qr −1 over an alphabet with q letters. Let 1, Lee and Manhattan metric. It is formed by the Minkowski r − i C 1 i q , be the ith translate of 1, where = 1. We lattice [37] with the generator matrix ≤ ≤ C C1 C construct the following code ∗: C 1 2 3 −  2 3 1  . − def it 3 1 2 ∗ = (xi1 , xi2 ,...,xiℓ ): xit 1 , (i1,i2,...,iℓ) 2  −  C { ∈C ∈C } The related anticode in this case is S . 3′ ,2 Theorem 10: The code rs∗ is a q-ary perfect single-error- Cq 1 correcting code of length q −1 . − F. A doubling construction for binary perfect codes rProof:rs Clearly,rs the length of the codewords from ∗ q 1 q 1 q 1 C is q −1 qr −1 = q −1 . We will first prove that the min- Let be a binary perfect single-error-correcting code − − − C r imum distance of ∗ is three. Let (xi1 , xi2 ,...,xiℓ ) and of length n = 2 1 and let e its extended code of C − C r y ,y ,...,y be two distinct codewords of . We now length 2r. Let 1 = , 2,..., 2 be its 2r translates ( j1 j2 jℓ ) ∗ r C 1 2C C2 C C r distinguish between two cases. and e = e, e ,..., e be the 2 even translates of its C C C C Case 1: If (i1,i2,...,iℓ) = (j1, j2,...,jℓ) then extended code. Let ∗ be the code constructed as follows: 6 C dH ((i ,i ,...,iℓ), (j , j ,...,jℓ)) 3 since 1 2 1 2 ≥ (xi1 , xi2 ,...,xiℓ ), (yj1 ,yj2 ,...,yjℓ ) 2 and dH ( 2)=3. def i i r ∈C C ∗ = (x, y): x e, y , 1 i 2 W.l.o.g., we can assume that i = j , i = j , and i = j , C { ∈C ∈C ≤ ≤ } 1 6 1 2 6 2 3 6 3 and hence xi1 = yj1 , xi2 = yj2 , and xi3 = yj3 , which It was proved by Phelps [20] that 6 6 6 implies that dH ((xi1 , xi2 ,...,xiℓ ), (yj1 ,yj2 ,...,yjℓ )) 3. Theorem 8: ∗ is a perfect single-error-correcting code ≥ Case 2: If i ,i ,...,i j , j ,...,j then there of length 2r+1 C 1. ( 1 2 ℓ) = ( 1 2 ℓ) − exists a t, 1 t ℓ, such that xit = yjt and It is readily verified that the perfect extended code e∗ is ≤ ≤it jt 6 it C since it = jt (i.e. 1 = 1 ), xit ,yjt 1 , and the code defined by it C C ∈ C dH ( )=3, it follows that d(xit ,yjt ) 3 and therefore C1 ≥ i i r d((xi1 , xi2 ,...,xiℓ ), (yj1 ,yj2 ,...,yjℓ )) 3. (x, y): x e, y e, 1 i 2 ≥ { ∈C ∈C ≤ ≤ } Thus, d ( ) 3. H ∗ n C ≥ q n r Let be a binary perfect single-error-correcting code of By Corollary 1, the size of 1 is = q − and B r 1+(q 1)n length n = 2 1 and let e its extended code of length rℓ C − r q rℓ rs r 1 − 2 B2 r the size of 2 is 1+(qr 1)ℓ = q − . Clearly, ∗ = 2 2 . Let e = e, e ,..., e be the 2 even translates of C − nℓ |C | |C | · B B B B r ℓ rℓ rs (n r)ℓ nℓ rs q its extended code. Let π = (π(1) = 1, π(2),...,π(2 )) be a 1 = q − q − = q − = 1+(q 1)nℓ . r |C | − permutation of 1, 2,..., 2 . The following theorem is an This implies by Corollary 2 that ∗rs is a q-ary perfect { } Cq 1 immediate consequence single-error-correcting code of length q −1 . − 7

IV. CONSTRUCTIONS FOR PERFECT AND DIAMETER Theorem 13: Let C1 and C2 be diameter perfect codes of PERFECT LEE CODES length n with minimum Manhattan distance four. Each code In this section we will modify the two product con- has 4n translates from which 2n are even translates. Let C1 C C2 C2n structions for perfect codes and diameter perfect codes in i = i, i ,..., i , i =1, 2, be these 2n even translates. the Hamming scheme to obtain perfect codes and diameter Let π = (π(1) = 1, π(2),...,π(2n)) be a permutation of 1, 2,..., 2n . The code C∗ defined by perfect codes in the Lee and the Manhattan metrics. { } C def Ci Cπ(i) ∗ = (x, y): x 1, y 2 , 1 i 2n (4) A. Diameter perfect codes with minimum distance four { ∈ ∈ ≤ ≤ } is a diameter perfect Manhattan code of length 2n, over Z, Let C and C be (n, 4, 4n,m) diameter perfect codes. 1 2 with minimum Manhattan distance four. Each code has 4n translates from which 2n are even C1 C C2 C2n translates. Let r = r, r,..., r , r = 1, 2, be these 2n even translates. Let π = (π(1) = 1, π(2),...,π(2n)) be B. Perfect single-error-correcting Lee codes a permutation of 1, 2,..., 2n . The following theorem is a { } In this subsection we use a modification of the product modification of Theorem 9. construction of subsection III to construct perfect single- C Theorem 11: The code ∗ defined by error-correcting Lee codes. def π(i) Let C be a perfect single-error-correcting Lee code of C Ci C 1 r ∗ = (x, y): x 1, y 2 , 1 i 2n q 1 { ∈ ∈ ≤ ≤ } length n = 2− , q odd, over an alphabet with τ(2n + 1) r is an diameter perfect Lee code of length 2n, over Zm, with letters, which has a total of q translates (the size of minimum Lee distance four. the Lee sphere), including C1 itself. Let πt = (πt(1) = C mn Proof: The size of the code r, r = 1, 2, is 4n and 1, πt(2),...,πt(2n)), 1 t ℓ, be a permutation of m2n m2n r ≤ ≤ hence the size of C is 2n 2 = . The Lee distance 1, 2,...,q . We have ℓ permutation, where the first per- ∗ 16n 8n { } of the code is four as the one given in Theorem 9. Since mutation is the identity permutation. Let 2 be a perfectrs C q 1 no proof is given in subsection II-F we will now present a single-error-correcting Hamming code of length ℓ = qr −1 r Ci r − proof. Let (x1, x2) and (y1,y2) be two distinct codewords in over an alphabet with q letters. Let 1, 1 i q , be i j 1 ≤ ≤ C∗ such that x C and y C . We distinguish between the ith translate of C1, where C = C1. We construct the 1 ∈ 1 1 ∈ 1 1 two cases. following code ∗ j π(i) π(j) C Case 1: If i = j then Ci = C and C = C . 6 1 6 1 2 6 2 Hence dL(x1,y1) 2, dL(x2,y2) 2, which implies that C Cπt(it) ≥ ≥ ∗ = (xi1 ,...,xiℓ ): xit 1 , (i1,...,iℓ) 2 dL((x1, x2), (y1,y2)) 4. { ∈ ∈C } ≥ Ci Case 2: If i = j then x1,y1 1 and x2,y2 Theorem 14: The code C∗ is a perfect single-error- π(i) ∈ ∈ rs C . Since x = y or x = y we have that correcting Lee code of length q 1 over an alphabet of 2 1 6 1 2 6 2 2− dL(x1,y1) 4 or dL(x2,y2) 4, respectively. Hence, size τ(2n + 1). ≥ ≥ dL((x , x ), (y ,y )) 4. C 1 2 1 2 rProof:rs Clearly,rs the length of the codewords from ∗ ≥ q 1 q 1 q 1 Thus d (C ) 4. r C L ∗ is 2− q −1 = 2− . The proof that the code ∗ has ≥ − The code C∗ is defined over Zm and hence the size of its minimum Lee distance three is identical to the related proof space is m2n. Since by (3) the size of the related anticode in Theorem 10. rℓ with diameter four is 8n and the minimum Lee distance of q rℓ rs By Corollary 1, the size of 2 is 1+(qr 1)ℓ = q − . C C n nC 1 − ∗ is four, it follows by Lemma 2 that ∗ is a diameter The size of C1 is τ (2n + 1) − . Clearly, C∗ = 2 perfect Lee code of length n, over Z , with minimum Lee ℓ rℓ rs nℓ (n 1)ℓ rℓ rs |nℓ r| (n 1)|Cℓ | · 2 m C1 = q − τ (2n + 1) − = q − τ q − = nℓ nℓ distance four. | rnℓ| rs nℓ τ (2n+1) r q − τ = qrs . Corollary 7: Let n = 2 p, where p is an odd prime C This implies by Corollary 4 that rs∗ is a perfect single- greater than one. There exists an (n, 4, 4n, 4p) diameter q 1 error-correcting Lee code of length 2− over an alphabet perfect code. of size τ(2n + 1). The lattice of the following generator matrix 2 2 V. ONTHE NUMBER OF PERFECT CODES AND , NON-PERIODIC CODESINTHE MANHATTAN METRIC  0 4  In this section we consider two problems related to form a (2, 4, 8, 4) diameter perfect code. By applying The- perfect codes. The first one is the number of nonequivalent orem 11 iteratively we obtain the following theorem. perfect codes, which was considered in many papers for Theorem 12: For each n = 2r, r 1, there exists an the Hamming schemes, e.g. [22], [26]. It was proved that (n, 4, 4n, 4) diameter perfect code. ≥ the number of nonequivalent perfect single-error-correcting cn Codes with the same parameters as in Theorem 12 were code of length n over GF(q) is qq , where c is a constant, generated by Krotov [38]. 0 < c< 1. The second problem is whether there exist Theorem 11 can be modified and applied on codes in Zn non-periodic perfect and diameter perfect codes in Zn with with the Manhattan metric. the Manhattan metric. Periodic and non-periodic codes were 8 mentioned in [12], but for the best of our knowledge no S is perfectly ρ-covered by C if for each element s S construction of non-periodic codes was known until recently. there is a unique element c C such that d(s,c) ρ. ∈ We note that we ask the question on non-periodic codes Theorem 15: There is no∈ finite subset of Zn≤that can in the Manhattan metric. The same question can be asked be perfectly ρ-covered by two different codes, using the for the Lee metric; the answer seems to be even more Manhattan metric. complicated than the one for the Manhattan metric. Proof: Assume the contrary; let S be a subset of Zn of We will not go into the more complicated exact com- smallest possible size, such that there exist two codes and C1 putations on the number of nonequivalent perfect and di- which ρ-cover S. Let k be the largest integer for which C2 ameter perfect codes. We will just count the number of there exists a codeword in or with the value k in one C1 C2 different perfect codes. The two product constructions given of the n coordinates. W.l.o.g. we can assume that such a in Section IV can be used to provide a lower bound on codeword is (k, x ,...,xn) . This codeword covers the 2 ∈C1 the number of perfect codes and diameter perfect codes in word (k + ρ, x ,...,xn) S. Since k can be the largest 2 ∈ the Lee metric. Given a diameter perfect code of length p, integer in a codeword of then the only codeword of C2 C2 p prime, the size of the related anticode is 4p, and hence which can cover (k + ρ, x2,...,xn) S is (k, x2,...,xn). ∈ there are 2p even translates. Therefore, there are (2p 1)! Thus, 1 (k, x2,...,xn) and 2 (k, x2,...,xn) − C \{ } C \{ } different perfect codes of length 2p, given in the construction perfectly ρ-cover a subset S′ S, a contradiction to the ⊂ of Theorem 11. Continue iteratively with the construction of assumption that S is such subset with the smallest size. Thus, Theorem 11 we obtain there is no finite subset in Zn that can be perfectly ρ-covered r by two different codes. r−i r (2ip 1)!2 For each n = 2 , r 1, there exists a non-periodic − ≥ Yi=1 diameter perfect code. If n =2 then such a code exists for each even minimum Manhattan distance 2R +2, 1 R. We r Z ≤ different diameter perfect codes of length 2 p over 4p. are using interleaving of the R +1 lattices used to construct Similarly, a bound on the number of perfect Lee code can diameter perfect codes with minimum Manhattan distance be obtained from the construction of subsection IV-B. In this 2R + 2. For a given R 1, let S = si i∞= be an ≥ { } −∞ computation we can also take into account the bounds on infinite sequence, where si ZR+1. Given S we construct the number of perfect codes in the Hamming scheme which the following set T of points∈ are used in the construction. def T = (2(R+1)i+(R+1)j+si, (R+1)j+si): si S, j Z Before we continue to discuss the topic of non-periodic { ∈ ∈ } Zn perfect codes and diameter perfect codes in we consider The sequence S will be called non-periodic if there are no a question related to both problems discussed in this section. nonzero integer ρ and an integer τ such that si = si+τ + Let C and C be two distinct subcodes of , such that 1 2 U ρ (mod R + 1). any elements in is within Manhattan distance one from a T U Theorem 16: If the points of the set are taken as centers unique codeword of C1 and a unique codeword of C2. If we for the spheres S2′ ,R then we obtain a tiling. The tiling is consider anticodes instead of spheres then the elements of non-periodic if the sequence S is non-periodic. C1 and C2 are centers of the anticodes which form a tiling Proof: First we note that the set of centers ((R + { of . If C1 is contained in a perfect single-error-correcting 1)j, (R + 1)j): j Z forms a connected nonintersecting ManhattanU code (or a diameter perfect code with minimum ∈ } diagonal strip with the spheres S2′ ,R. The same is true for Manhattan distance four) C then it can be replaced by the set of centers (2(R + 1)i + (R + 1)j + si, (R + 1)j + C to obtain a new different perfect single-error-correcting { 2 si): j Z for any given i Z. The set of centers Manhattan code (or a diameter perfect code with minimum (2(R + 1)∈i +} (R + 1)j, (R + 1)j∈): i Z, j Z is { ∈ ∈ } Manhattan distance four) C′. The technique was used in the exactly the lattice formed from the generator matrix Hamming scheme to form a large number of nonequivalent R +1 R +1 perfect codes [25], [26]. Same technique can be considered , for the Lee metric.  2(R +1) 0  The constructions in Section IV can provide such a space which forms a diameter perfect code with minimum distance and codes C1 and C2. If we change the permutation of 2(R + 1). Finally, replacing the set of centers (2(R + 1)i + Uthe last coordinate by one transposition, it is easily verified (R + 1)j, (R + 1)j): j Z in this tiling{ by the set of ∈ } the intersection of the two generated codes is relatively large centers (2(R+1)i+(R+1)j +si, (R+1)j +si): j Z η 2 { ∈ } (usually about −η of the code size, where η =2n for the is just a shift of the diagonal strip in a 45 degrees diagonal construction of the diameter perfect codes and η = qr for direction. This does not affect the fact that the set of these the construction of the perfect single-error-correcting code). centers forms a tiling with the spheres S2′ ,R. The two parts which are not in the intersection will have It is readily verified that if the sequence S is non-periodic the role of C1 and C2. These can be the building block for then also the tiling generated from T is non-periodic. non-periodic codes in the Manhattan metric. This is left as The construction of two-dimensional non-periodic diameter a problem for future research. Unfortunately, such a finite perfect codes yields an uncountable number of diameter space Zn and codes C and C cannot exist. A subset perfect codes. The number of nonequivalent perfect codes U ⊂ 1 2 9 formed in this way is equal the number of infinite sequences anticodes and perfect codes. During an ITW meeting in over the alphabet ZR+1. Ireland at the beginning of September 2010 he brought to Finally, it is easy to verify the following theorem. his attention that the size of a maximum anticode in the Theorem 17: Let C1 be a non-periodic diameter perfect Manhattan metric was found recently. 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