Quasi-perfect Lee Codes of Radius 2 and Arbitrarily Large Dimension

A construction of 2-quasi-perfect Lee codes is given over the space $\mathbb Z_p^n$ for $p$ prime, $p\equiv \pm 5\pmod{12}$ and $n=2[\frac{p}{4}]$. It is known that there are infinite such primes. Perfect codes for the Lee-metric were conjectured by Golomb and Welch not to exist, which has been proved for large radii and also for low dimension. The codes found are very close to be perfect, which tells about the nature of the conjecture. Some computations show that related graphs are Ramanujan, which could provide further connections between the fields.


Introduction
Golomb and Welch conjectured in their seminal paper [13] that perfect Lee codes do only exist for spheres of radius r = 1 or in Lee spaces of dimension n = 1, 2. A constructive result for 1-perfect Lee codes was also given in that paper. Moreover, for a radius sufficiently greater than the space dimension, a negative existence result was given by approximating the problem to the densest tilling of R n with cross-polytopes. Afterwards, Molnár enumerated all lattice-like 1-perfect codes in [28]. Later, Post in [30] gave a strong negative result. If a perfect code exists, Post determined an upper bound for its radius, in terms of the dimension, specifically, the ra- * C. Camarero and C. Martínez  √ 2 − 1 2 for n ≥ 6. Later, J. Astola [7] and Lepistö [22] improved the bound given by Post to a quadratic relation between r and n, which can be considered as an Elias-type bound for Lee codes. Those negative results in [30,7,22], suggest that the conjecture is more difficult for radius 2, as argued by Horak in [16].
Other authors have considered the conjecture for small dimensions. For example, Gravier et al. in [14] proved the non-existence of perfect codes in 3-dimensional Lee spaces, even considering spheres of different radii. Dimension 4 was considered by Spacapan in [34], again with the possibility of spheres of different radii but at least 2. Also, Horak in [17] and [16] proved the non-existence of perfect Lee codes in spaces of dimension up to n ≤ 6. Later, to achieve higher dimensions, Horak and Grošek in [19] restricted the problem to linear codes and verified computationally the non-existence of perfect Lee codes for dimension n ≤ 12 and radius r = 2.
On the other hand, several papers have considered problems around the conjecture that could give some insight. One approach has been to generalize the Lee metric. Huber in [20] gave 1-perfect codes over Gaussian integers and some non-perfect codes with greater correction. In [9] Costa et al. considered a relation between tessellations, graphs and codes over flat tori. In [24,26,25] Martinez et al. gave a generalization of the Lee distance by means of a family of Cayley graphs over Cayley-Dickson algebras. Also, the existence of perfect codes being ideals of the algebras was considered. Nishimura and Hiramatsu in [29] generalized the Lee distance using a surjective function from Z l into a finite field and constructed some non-perfect 2-error correcting codes for this metric.
The existence of Lee codes has also been considered in terms of the size q of the alphabet. AlBdaiwi et al. in [3] enumerated all the alphabet sizes q such that there exists a linear 1-perfect Lee code over Z n . In [6] H. Astola and Tabus obtained, for small alphabet size q and dimension n, an upper bound of the number of codewords of error correcting Lee codes.
Recently, a new approach has been taken in terms of diameter perfect codes, which were introduced by Ahlswede et al. in [1]. A subset C ⊆ Z n q is a diameter perfect code if there exists an anticode A such that |C||A| = q n . This concept generalizes perfect codes since diameter perfect codes with minimum distance being odd are in fact the perfect codes. Etzion in [11] built diameter perfect codes of minimum distance 4. Later, Horak and AlBdaiwi [18] enumerated the arities q such that there are 4-diameter perfect codes over Z q . Araujo et al. in [4] presented a generalization of diameter perfect Lee codes, together with a new conjecture that extends the conjecture by Golomb and Welch. Etzion et al. in [12] built Lee codes for large dimension by means of weighing matrices.
A different approach to the existence of perfect Lee codes has been to relax de condition of being perfect. Thus, quasi-perfect codes for the Lee metric have been considered just in a small number of papers. AlBdaiwi and Bose in [2] presented some quasi-perfect codes for dimension n = 2. Also, in [19] the authors presented some quasi-perfect codes for n = 3 and a few radii. Later, Queiroz et al. in [31] characterized quasi-perfect codes over Gaussian and Eisenstein-Jacobi integers being linear, as a consequence linear quasi-perfect Lee codes are obtained for n = 2.
In the present paper a explicit construction of linear quasi-perfect Lee codes of radius 2 for arbitrarily large dimension is given. As it will be shown, these codes are very close to be perfect since they have half of the density of potential perfect codes. In the authors opinion, the existence of these quasi-perfect codes, hints that maybe a perfect code could exist for low radius; and if they do not exist then the proof must be of a very different nature than the proofs in previous papers dealing with the conjecture.
These quasi-perfect 2-error correcting Lee codes will be defined by means of Cayley graphs over Abelian finite groups. The degree of the graph will be the double of the dimension of the Lee space. The order of the graph will be in inverse relation to the density of the quasi-perfect code. Thus, the main contribution of the paper is presented in the next result.
Theorem 1. For any prime p ≥ 7 such that p ≡ ±5 (mod 12) there exists a linear 2-quasi-perfect p-ary Lee code over Z n p , where n = 2 p 4 and with p n−2 codewords.
Note that the notation [a] stands for the closest integer to the rational number a. As an example of the codes obtained in previous result, let us consider the following: Example 1. Let n = 4, p = 7. Then, the code over Z 4 7 defined by the parity-check matrix results in a 2-quasi-perfect 7-ary Lee code over Z 4 7 . This code has p n−2 = 49 codewords. It is known that perfect codes do not exist in this case since the sphere packing bound is 7 4 41 ≈ 58.56. As a consequence of Dirichlet's theorem on arithmetic progressions, there are infinite primes p such that p ≡ 5 (mod 12) and infinite primes such that p ≡ −5 (mod 12). Thus, for any constant c, there is a prime p ≡ ±5 (mod 12) such that the dimension n = 2 p 4 is greater than c. As a consequence of this and Theorem 1, it is obtained that: There are infinite n ∈ N such that there exists a 2-quasi-perfect Lee code over a ndimensional Lee space.
As it will be seen later, the result is constructive, and any application that requires the use of Lee-codes can benefit from it. For example, Roth and Siegel in [32] considered BCH Lee codes and their application to constrained and partial-response channels. Using space embeddings, Jiang et al. in [21] gave a method to construct Charge-Constrained Rank-Modulation codes (CCRM codes) from Lee error-correcting codes, which could be employed for flash memories. H. Astola and Stankovic in [5] considered Lee codes to build decision diagrams.
The rest of the paper is organized as follows. Since the codes considered in this paper will be defined by means of Cayley graphs, in Section 2 the relation between Lee codes and Cayley graphs over Gaussian integers is stated. Moreover, the family of Cayley graphs under study is defined. Then, in Section 3 the Cayley graphs selected are proved to have error correction capacity 2. In Section 4 those Cayley graphs are shown to attain diameter 3, which concludes that they define 2-quasi-perfect codes. Finally, in Section 5 the results presented in this paper are discussed, and some open problems and future lines of research are detailed.

Codes and Graphs
Linear 2-quasi-perfect p-ary linear Lee codes are going to be defined by means of Cayley graphs. Therefore, the correspondence between a linear code and a Cayley graph is explained in this section. Moreover, some fundamental definitions are stated here.
Since Lee codes are the target of our study, the natural space to be considered is the finite integer lattice Z n p . However, for convenience, also the infinite lattice Z n will be considered. Therefore, a code C will be a subset of either Z n p or Z n . This code is said to be linear or lattice-like if it is a subgroup of the corresponding space.
In the space Z n it will be used the Manhattan distance. For any two words v, w ∈ Z n its Manhattan distance is defined as: On the other hand, the Lee distance will be the metric when considering Z n p . For v, w ∈ Z n p its Lee distance is defined as Since the Lee distance becomes the Manhattan distance for p = ∞, there will be no opportunity for confusion. In both cases the weight of a word v is defined as its distance to the zero vector, which will be denoted as |v| = d(v, 0). For any positive integer r, the Lee sphere of radius r is defined as all the points with weight less or equal to r, that is: Note that, for any dimension n ≥ 1, the cardinal |B n 2 | = 2n 2 + 2n + 1, [13]. A code C is said t-error correcting if t is the greatest integer such that for any word w there is at most one codeword c ∈ C with d(w, c) ≤ t. A code C is said r-covering if r is the smallest integer such that for any word w there is at least one codeword c ∈ C with d(w, c) ≤ r. Then, a code that is both t-error correcting and t-covering is said to be perfect. Golomb and Welch in [13] conjectured that there only exist perfect Lee codes for t = 1 or n = 2. Therefore, the existence of quasi-perfect codes must be studied since they are the best alternative to the perfect ones. Thus, a code that is t-error correcting and (t + 1)-covering is said to be t-quasi-perfect. In this work 2-quasi-perfect Lee codes are found for arbitrarily large dimension. This is done by the construction of a family of Cayley graphs that leads to the codes definition. In the rest of the section, this relation between codes and graphs, using tilings, is stated. For simplicity, in the remaining section the infinite lattice Z n will be considered. An equivalent relation can be stated in the case of Z n p . Codes and tilings are closely related as it is manifested in many papers. Given a linear code C, the tessellation induced by the Voronoi regions of the codewords can be considered. The Voronoi region of a codeword is composed by the words that are closer to it than to other codewords. Since the code is linear, the tessellation is congruent, that is, all the tiles have the same shape and size. If C is a t-perfect error correcting code then the tiles obtained are translations of the Lee sphere of radius t. Otherwise, for a t-correcting and r-covering code, each induced tile contains a Lee sphere of radius t and it is contained in a Lee sphere of radius r. Reciprocally, the centers of the congruent tiles of a lattice-like tessellation can be used to define a linear code.
Once the tessellation induced by the linear code is obtained, then this tessellation can be used to define a Cayley graph. Given a group Γ and a set of generators H = {β 1 , . . . , β s } ⊂ Γ, the Cayley graph over Γ generated by H is defined as the graph with set of vertices the elements of Γ, and adjacencies (u, u+β i ), for every u ∈ Γ and i = 1, . . . , s. Now, if H = −H then the Cayley graph is undirected, as the ones considered in this paper. Thus, this graph is denoted by Cay(Γ, H). The distance between two vertices in a graph is calculated as the length of a minimum path that joins them. Therefore, the diameter of a graph is the maximum among distances between every pair of vertices. Now, let us consider a tessellation of congruent tiles by translations. Then, a graph can be defined by considering the words inside the tile centered at codeword 0 as its set of vertices. To define the adjacency, two different situations can be considered. For the vertices inside the tile, two vertices are adjacent if they are at a distance 1. In the case of vertices in the boundary of the tile, two vertices u and v are adjacent if there is a tile center c such that d(u, c+v) = 1. Note that, since the tessellation comes from a linear code, the graph is a undirected Cayley graph over the Abelian group Z n /C. Now, if t is the greatest integer such that the Lee sphere B n t is contained in the tile, then this graph has |B n t | vertices at distance t or less. By analogy to the concept of correction in codes, this value t will be referred as the error correction capacity of the graph. If r is the smallest positive integer such that B n r contains the tile, then the graph has diameter r. Given a Cayley graph it is straightforward to obtain a lattice-like congruent tilling. The tile can be defined by the representation in minimum distances of the set of vertices. Then, the tiling of the space is induced by the adjacency of the peripheral vertices.
In the rest of the paper a family of Cayley graphs over Gaussian integers will be considered. Let us denote by Z[i] = {a + bi | a, b ∈ Z} the ring of the Gaussian integers. In Appendix A there is a summary of some fundamentals on this ring. Given an integer prime p, let us denote by Z[i]/pZ[i] the quo-tient additive group of the Gaussian integers over the group generated by (p) ⊂ Z[i]. Thus, the graph is defined as follows.
Definition 3. Given an integer prime p, let us define the Cayley graph denotes the norm of β. Moreover, the adjacency in the graph is determined by the elements with unitary norm. In the following sections, it will be proved that G p induces a 2-quasi-perfect Lee code over Z n p under some conditions. Therefore, it must be determined which primes p are such that G p has error correction capacity 2 and diameter 3.

Error Correction Capacity of G p
As explained in previous section, 2-quasi-perfect Lee codes are going to be obtained by means of Cayley graphs. In particular, it will be determined under which conditions the Cayley graph G p over the additive group Z[i]/pZ[i] and generating set those elements with unitary norm induces a 2-quasi-perfect code. In this section it will be proved that p ≡ ±5 (mod 12) implies that G p has error correction capacity 2 over Z n p for n = 2[ p 4 ]. Hence, in the remainder of the paper, let us assume that p > 2 is a prime integer. Therefore, the natural number n = 2[ p 4 ] fulfills p = 2n ± 1.
First, let us introduce some notation. Given a stand for its real part and ℑ(β) = b 2 for its imaginary part. Then, the following formula about the norm of a sum of Gaussian integers will be useful in several points of this paper.
Then, the previous result can be used to prove the following technical lemma: As a consequence, y 2 = z 2 and therefore y ∈ {±z}, which means γ 1 ∈ {γ 2 , γ * 2 }.
) the multiplicative group formed by the units of the ring. Then, the set is clearly a multiplicative normal subgroup of G. It is actually a cyclic group, although this fact will not be used in the proofs. Note that H is the set of adjacencies of G p , this is, G = Cay(Z[i]/pZ[i], H). For any γ ∈ Z[i]/pZ[i], the following notation is introduced: Notice that if γ ∈ G, then γH is the coset of H in G with respect to γ. Nevertheless, this notation is also defined for elements outside G, i.e., for zero divisors The following lemma tells us that cosets can be identified by the norms of its elements.
Proof. In order to prove the sets equality, it will be first proved that γH ⊆ {β ∈ G | N (β) = N (γ)}. Thus, let us consider β ∈ γH and it has to be proved that N (β) = N (γ). Since β ∈ γH, then there exists η ∈ H such that β = γη. Hence N (β) = N (γ)N (η) = N (γ). Now, let us consider the other inclusion, that is, Theorem 9 states that the degree of the graph G p is 2n. To prove it some particular cases of the Quadratic Reciprocity Law will be necessary, which are recalled in the following theorem for selfcontainedness.
Theorem 8 (Quadratic Reciprocity). If p is an integer prime, then: 1. The number of solutions to −1 = x 2 in Z/pZ is: 2. The number of solutions to 3 = x 2 in Z/pZ is: Proof. It is clear that Therefore, let us consider the solutions of x, y ∈ Z/pZ of equation x 2 + y 2 = 1. First, if x = 1 then y 2 = 0 whose unique solution is y = 0. Let us assume x = 1 to look for the rest of solutions. Since x = 1, x − 1 has inverse and it is possible to define s = y/(x − 1) ∈ Z/pZ. By considering the intersection of the straight line y = s(x − 1) with the curve x 2 + y 2 = 1 it is obtained that x 2 + (s(x − 1)) 2 = 1. Finally, just to ensure that the counted solutions are all different, note that if for a pair s 1 , s 2 the same solution (x, y) is obtained, then s 1 = s 2 = y/(x − 1).
Next, it can be easily obtained the following consequence of previous theorem, which will be used in Section 4 to determine the diameter of the graph G p .
Before stating the conditions under which G p has error correction capacity 2, the following lemma is going to be proved. This lemma determines the number of possible norms among the neighbours of a vertex with a given norm. For the case of c being a square non-residue let us proceed in a similar way. It is obtained that N (1+β 2 ), . . . , N (1+β n−1 )}, which are different by Lemma 5. Hence |N p (c)| = n.
As it will be noted afterwards, the case c = 1 in previous lemma is going to be used to prove the error correction capacity. Later, the fact that n is a lower bound of N p (c) will be considered to determine the graph diameter.
To finish the section, next theorem establishes the conditions for p such that G p has error correction capacity 2.
Theorem 12. Let p be a prime integer satisfying p ≡ ±5 (mod 12). Let n = 2 p 4 . Then, the Cayley graph G p has error correction capacity 2.
Proof. As it was explained in previous section, it has to be proved that G p contains |B n 2 | = 2n 2 + 2n + 1 vertices at distance 2 or less from 0. Clearly, 0 is the unique vertex at distance 0. Now, the set H contains all the vertices at distance 1 and |H| = 2n by Theorem 9. {0}). Thus, let us prove that |A| = 2n 2 . By Lemma 7 and Corollary 10, |A| = 2n · |N p (1) \ {0, 1}|. Since 1 is always a square residue for any p, hence by Lemma 11, |N p (1) \ {0}| = n. It remains to be proved that 1 does not belong to N p (1).
Remark 13. If p is a prime greater than 3 that does not satisfy p ≡ ±5 (mod 12), then p ≡ ±1 (mod 12). In this case, G p only contains 2n 2 + 1 vertices at distance 2 or less from vertex 0. Although it is not a 2-error correcting code, it is very close since only 2n syndromes cannot be corrected.

Diameter of G p
In this section it will be proved that G p has diameter 3 for any prime p > 5. The proof will be separated into two subsections, the first one considering the case p ≡ 3 (mod 4) and the second one the case p ≡ 1 (mod 4). Also, from here onwards it will be assumed again that n = 2[ p 4 ]. Note that, since |Z[i]/pZ[i]| = p 2 > |B n 2 |, there are vertices outside the sphere of radius 2, which means that the diameter of the graph is at least 3. As it will be seen next, the proofs proceed by reductio ad absurdum by the assumption of the existence of a vertex at a distance 4 from vertex 0, thus reaching a contradiction.

Case p ≡ 3 (mod 4)
In this case the proof of the diameter can be easily obtained by using a counting argument. Note that in this case p = 2n − 1 and therefore Z[i]/pZ[i] is a field.
Theorem 14. For any prime p such that p ≡ 3 (mod 4) the graph G p has diameter 3.
Proof. By reductio ad absurdum let us assume that there exists a vertex γ ∈ Z[i]/pZ[i] at distance 4 of vertex 0. Let c = N (γ). Since γ is so far, it is obtained that N p (1) ∩ N p (c) = ∅.
Let us denote by W t (0) the number of vertices at a distance t from vertex 0. Then, {W t (0) | t = 0, . . . , 4} is the distance distribution of the graph G p . Now, the cardinals W 1 (0) = |H| and W 4 (0) ≥ |γH| can be calculated by Corollary 10. Also, by Lemma 11 it can be computed that |N p (1)| = n + 1 and |N p (c)| ≥ n. Thus, the bounds for the distance distribution obtained are summarized next: As a consequence, the total number of vertices satis- Proof. On one hand, by Corollary 10, the cardinal |ζH| is 2n. On the other hand, |{xζ | x ∈ Z/pZ, x = 0}| has p − 1 = 2n elements. Since both sets have the same size, it is enough to prove one inclusion to show the sets equality. Therefore, let us prove the left to right inclusion.
Let β = a + bi be an element of norm 1 and ζ = u + vi a proper zero divisor, hence of norm 0. As ζ = 0 and Z/pZ is a field, both u and v are nonzero.
Finally, note that x cannot be zero, since it would imply that β were a zero divisor, contradicting N (β) = 1.
The following lemma has its inspiration in Lemma 11, but with the intention to generalize to the case of zero divisors and to give a stronger result. Proof. Let ζ = u + vi be a proper zero divisor. By Lemma 15 and by making few calculations, To finish, note that y = 1 + 2xu with x = 0 has solution for every value of y except 1.
The previous lemma indicates that proper zero divisors are neighbours of every vertex at distance 2 from 0, and hence they are at distance 3 from 0. Then, the following lemma gives a polynomial description of the sets N p (t).
Lemma 17. Let p ≡ 1 (mod 4) be a prime in Z. For any t ∈ Z/pZ, t = 0, it is obtained that Proof. By the first item of Theorem 8, there exists r ∈ Z/pZ such that r 2 = −1. Note that x −1 (x + 1)(x + t) = x + tx −1 + t + 1. First, let us prove the left to right inclusion of the sets. In this aim, let β = a + bi, N (β) = a 2 + b 2 = t for a generic element N (1 + β) in N p (t). Thus, let us check that x = a + rb Hence, For the right to the left inclusion, let x = 0 and which ends the proof.
The intersection between N p (1) and N p (t) will be given by the roots of polynomial P t (x, y) = y(x + 1) 2 − x(y + 1)(y + t). In order to apply the Hasse-Weil bound, the polynomial must be irreducible. Therefore, let us introduce the following definition and two useful results in Lemma 19 and Corollary 20.
Definition 18. Given a field F, a polynomial P ∈ F[x, y] is called absolutely irreducible if it is irreducible in the algebraic closure of F.
Proof. The polynomial P t (x, y) = xy(x − y) + (1 − t)xy + y − tx has degree 3. If P t (x, y) is not absolute irreducible, then there exist polynomials A(x, y), B(x, y) with coefficients in the algebraic closure of Z/pZ such that P t (x, y) = AB with deg A(x, y) = 2 and deg B(x, y) = 1. Furthermore, the product of the leading terms of A(x, y) and B(x, y) must be xy(x − y). Let us consider the following three mutually exclusive cases, depending on polynomials A(x, y) and B(x, y) is a and the one of y 2 is −b. By hypothesis, both are 0 in P t (x, y). Then, the coefficient of xy is d = 1 − t, the coefficient of x is c = −t and the coefficient of y is −c = t = 1. Hence, for t = 1 there exists the factorization P 1 (x, y) = (xy − 1)(x − y).

Case
A(x, y) = (x(x−y)+ax+by +c), B(x, y) = (y + d). Now, the coefficient of x 2 in A(x, y) · B(x, y) is d = 0 and the coefficient of y 2 is b = 0. Then, the coefficient of xy is a = 1 − t, the coefficient of x is 0 = −t and the coefficient of y is c = 1. Hence, for t = 0 there exists the factorization P 0 (x, y) = (x 2 − xy + x + 1)y.

Case
A(x, y) = (y(x−y)+ax+by +c), B(x, y) = (x + d). The coefficient of x 2 is a = 0 and the coefficient of y 2 is −d = 0. Then, the coefficient of y would be 0 = 1, which implies that there exists no factorization.
Finally, there are factorizations of P t (x, y) only for t = 0 and t = 1, which proves the result.
Proof. If h P t (x, y) had a factorization, then its evaluation at z = 1 would be a factorization of P t (x, y), contradicting Lemma 19.
Finally, let us conclude the section by proving the main result.
Theorem 21. If p is a prime such that p ≡ 1 (mod 4) and p > 5, then the diameter of G p is 3.
Proof. Let us proceed again by reductio ad absurdum. In this aim, let us assume the existence of a vertex γ at distance 4 from 0 in G p , with p fulfilling the hypothesis of the statement. Let t = N (γ). Note that t = 1 since vertices with norm equal to 1 are at distance 1. Also, t = 0 by Lemma 16. Hence, by Lemma 7, the vertices with norm in the set N p (t)\{0} are at distance at least 3. Meanwhile, the vertices with norm in N p (1) \ {0} are at distance at most 2 from 0. Therefore, the intersection of previous two sets is N p (1) ∩ N p (t) = {0}. Now, using polynomial notation, previous sets equality is equivalent, by Lemma 17, to the nonexistence of solutions to x −1 (x+ 1) 2 = y −1 (y + 1)(y + t) other than x = −1. Let us highlight that the solution x = −1 corresponds with norm 0. Thus, vertices in H have vertex 0 as their neighbour, while vertices in γH have as some of their neighbours vertices that are proper zero divisors.
The contradiction will be obtained when proving the existence of a solution to P t (x, y) = 0 other than the trivial ones (x, y) ∈ {(0, 0), (−1, −1), (−1, −t)}. In this aim, let us define the varieties where P 2 Z/pZ denotes the projective space of dimension 2 over Z/pZ. The notation (x : y : z) indicates a projective point, which is the same point as (λx : λy : λz) for any λ = 0. Thus, affine solutions can be recovered by taking λ = z −1 ; except for solutions (x : y : 0), which are the points at the infinite.
Remark 22. G 5 has diameter 4 since, vertex 2 + 2i and its associates are at distance 4 from vertex 0.

Discussion
In this final section conclusions of this work and future research are going to be presented. In the first subsection, the main result is rewritten using paritycheck matrices. Also, a formal proof of the infiniteness of the constructed family of quasi-perfect codes is given. Some considerations on the density of the codes are addressed. Moreover, other examples of codes presenting greater density and an upper error correction capacity are shown. In the final subsection, the authors aim to exhibit the connections of the graphs considered in the present study with other graph theoretical problems, trying to give a new insight into the perfect Lee codes conjecture formulated by Golomb and Welch more than forty years ago.

Quasi-perfect Lee codes
As it has been proved in previous Sections 3 and 4, G p has error correction capacity 2 and diameter 3, for any prime p > 5 and p ≡ ±5 (mod 12).
Dirichlet's theorem on arithmetic progressions asserts that in an arithmetic progression there are infinite primes. As a natural consequence, congruences can be considered as arithmetics progressions, and therefore it can be straightforwardly obtained the following: Corollary 23. There are infinite n ∈ N such that p = 2n ± 1, p ≥ 7 prime in Z, p ≡ ±5 (mod 12).
Then, when applying the previous result it is obtained: Corollary 24. The family of graphs G p contains infinite graphs with error correction capacity 2 and diameter 3. This can be verified by realizing that the word c is associated to vertex M c and that c belongs to the code if and only if it is associated to vertex 0; i.e., the codewords are exactly the words c such that M c = 0.
However, the code associated to this matrix belongs to the space (Z[i]/pZ[i]) n/2 . In order to obtain the Lee code over (Z/pZ) n and a parity-check matrix with integer entries, every β has to be substituted by Therefore, the parity-check matrix associated to the graph G p is . Now, let us give some considerations on the quality of the codes constructed. Note that, since the Lee sphere of radius 2 contains |B 2 | = 2n 2 + 2n+ 1 words, then the graph induced by any 2-quasi-perfect linear code has at least 2n 2 + 2n+ 1 vertices. The graphs G p constructed in this paper have p 2 vertices. Therefore, for the case p = 2n + 1, the number of vertices is p 2 = 4n 2 + 4n + 1 = 2|B 2 | − 1. Also, for the case p = 2n − 1, the number of vertices is p 2 = 4n 2 − 4n + 1 = 2|B 2 | − 8n − 1. Thus, the reached vertices are asymptotically the double of which would be reached in the graph associated to a perfect code. In other words, the density of the codes presented is 1 p 2 . Although the obtained density is quite good, for some small cases (low dimension), graphs with a smaller number of vertices have been computationally found. Let us consider the following examples. Example 3. Let n = 16 be the dimension. In this case, by extending the search to a different ring, a new graph has been found. The graph is build over the Quaternion integers modulo p = 5, being the generator set H = ±{1, 1 + 2i + 3j, 3i + 4j + 1k, 3 + 4i + 3j}. In this case, the number of vertices of the graph is p 4 = 625, which is 15% over |B 16 2 | = 545. The previous small examples suggest that there exist codes very close to be perfect, although general constructions appear to be difficult to find.
From the result by Post [30] it was obtained that there are not perfect codes with radius greater than the dimension of the space. Previously to that paper, Golomb and Welch [13] had already noted that there cannot be perfect codes with correction greater than a constant that depends on the dimension by the use of the maximum density of packing with cross-polytopes. Clearly, this can be applied to quasiperfect codes. For every n there exists t n such there are not t-quasi-perfect codes for t ≥ t n . Hence, this might suggest that radius 2 case is an exceptional one. Nevertheless, a few 3-quasi-perfect codes have been found for small dimensions. Note that in this case the n-dimensional sphere of radius 3 has cardinal |B n 3 | = can be seen, the first example is just 31% over the cardinal of the sphere, while the second and third are 79% and 102%, respectively. Any of the three examples can be considered as 3-quasi-perfect codes really near to the perfect one.
In the authors opinion, the construction of an infinite family of graphs containing these codes or similar ones would have a great value, both practical and in a better understanding of the Golomb and Welch conjecture.

Related Problems
Other interesting problems from different areas than from Coding Theory could be profited from this study. For example, this graph theoretical study of perfect codes can be seen as the reverse of the degreediameter problem for Cayley graphs over Abelian finite groups [27]. In this problem, for a given diameter, graphs with the maximum possible number of vertices are searched. Specifically, for a positive integer t, graphs providing t-covering codes but without caring about the correction are looked for. Note that in this case, the order of the graphs obtained is lower than the cardinal of the corresponding sphere |B n t |. Therefore, in the present paper graphs providing t-correcting codes and enforcing additionally (t + 1)-covering have been constructed. In our case, the order of the Cayley graphs is always greater than the cardinal of the sphere |B n t |. The degree-diameter problem for t = 2 and t = 3 has been considered in [23,33]. In that papers families of graphs with smaller number of vertices than the sphere cardinal were given. Specifically, one of the graph constructions in Macbeth et al. [23] is given for some infinite degrees 2n of graphs of diameter 2 and 3 2 (n 2 − 1) = 3 4 |B n 2 | − 3 2 n − 9 4 vertices. Then, Vetrík [33] constructs graphs with diameter 3 and 9 128 (2n + 3) 2 (2n − 5) vertices, which is asymptotically 27 64 |B n 3 |; it is remarkable that these graphs have error correction capacity 1 instead of the hoped 2, and thus they do not induce quasi-perfect codes. Note that a Cayley graph attaining the degree-diameter bound will induce a perfect code and vice versa.
Furthermore, the graphs considered in this paper seemed to be good expanders. Therefore, the authors computed the spectrum of some of them and the obtained values exhibit that they are Ramanujan graphs. Ramanujan graphs are good expander graphs that attain the spectral bound [10]. More specifically, G is a Ramanujan graph if and only if for every eigenvalue of its adjacency matrix λ it is hold either |λ| = deg(G) or |λ| ≤ 2 deg(G) − 1. Therefore, the following conjecture is proposed.
Conjecture 25. G p is a Ramanujan graph for any prime p ≡ 3 (mod 4). This conjecture has been verified for all primes p < 1000; the only primes in that range for which G p is not Ramanujan are 17, 53 and 541. Moreover, the authors believe that most primes fulfilling p ≡ 1 (mod 4) give also G p being Ramanujan graphs. Therefore, the proof of this conjecture and the study of the relation between Golomb and Welch conjecture and spectral analysis will be considered as future work.
is always p 2 . Then, the ring is a field if and only if p ≡ 3 (mod 4). The Gaussian norm becomes a function N : Z[i]/pZ[i] → Z/pZ; and Z/pZ is always a field. Finally, the norm of an element is 0 if and only if it is a zero divisor.