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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

Restriction conditions on PL(7, 2) codes (3 ≀ |𝓖i| ≀ 7)

Catarina N. Cruz / Ana M. ďAzevedo Breda
Published Online: 2018-04-02 | DOI:Β https://doi.org/10.1515/math-2018-0027

Abstract

The Golomb-Welch conjecture states that there is no perfect r-error correcting Lee code of word length n over β„€ for n β‰₯ 3 and r β‰₯ 2. This problem has received great attention due to its importance in applications in several areas beyond mathematics and computer sciences. Many results on this subject have been achieved, however the conjecture is only solved for some particular values of n and r, namely: 3 ≀ n ≀ 5 and r β‰₯ 2; n = 6 and r = 2. Here we give an important contribution for the case n = 7 and r = 2, establishing cardinality restrictions on codeword sets.

Keywords: Perfect Lee codes; Golomb-Welch conjecture; Space tilings

MSC 2010: 05B40; 05E99

1 Introduction

Problems involving space tilings are common in coding theory. In fact, special types of tilings can be regarded as error correcting codes which are essential on correct transmission of information over a noisy channel, see [1, 2].

In this paper we deal with tilings of β„€n by Lee spheres, where n is a positive integer number. The study of these tilings was introduced by Golomb and Welch, see [1, 3], where they related these tilings with error correcting codes considering the center of a Lee sphere as a codeword and the other elements of the sphere as words which are decoded by the central codeword. When a Lee sphere of radius r tiles the n-dimensional space, the set of all centers of the Lee spheres, that is, the set of all codewords, produces a perfect r-error correcting Lee code of word length n, which will be denoted by PL(n, r) code. The interest in Lee codes has been increasing due to their several applications, see, for instance, [4, 5, 6, 7].

The question β€œfor what values of n and r does the n-dimensional Lee sphere of radius r tile a n-dimensional space?” was formulated by Golomb and Welch in [1], where they proved: (i) n-dimensional Lee sphere of radius 1 tiles the n-dimensional space for any positive integer n; (ii) for each r β‰₯ 1, there exists a tiling of the n-dimensional space by Lee spheres of radius r for n = 1, 2. In other words, there exist PL(n, 1), PL(1, r) and PL(2, r) codes for any positive integer numbers n and r, respectively. These codes have been extensively studied by other authors, see, for instance, Stein and SzabΓ³ [8].

According to Golomb and Welch, it seems that there is no PL(n, r) code for other values of n and r, that is:

Conjecture (Golomb-Welch)

There is no PL(n, r) code for n β‰₯ 3 and r β‰₯ 2.

There exists an extensive literature on the subject, however the Golomb-Welch conjecture is still far from being solved. Actually, the conjecture is proved for 3 ≀ n ≀ 5 and r β‰₯ 2, see [9, 10, 11], and for n = 6 and r = 2, see [12]. The difficulty to prove the conjecture has led some authors to consider special types of PL(n, r) codes, such as linear and periodic ones, see [13, 14, 15]. It should be pointed out that Horak and Grosek, in [13], have proved, using a new approach, the nonexistence of linear PL(n, 2) codes for 7 ≀ n ≀ 12.

As stated previously, a Lee sphere of radius 1 tiles the n-dimensional space for any positive integer n. It seems that the most difficult cases of the Golomb-Welch conjecture are those in which r = 2. Following an intuitive and geometric reasoning, it seems that the bigger is the radius of the Lee sphere the more difficult is to tile the space with this sphere.

Here we will give a contribution for the case n = 7 and r = 2 presenting a possible strategy to prove the non-existence of PL(7, 2) codes. We believe that this strategy will allow us, in the future, to get the proof of the non-existence of such codes. Our strategy does not use computational methods and is faithful to the geometric idea of the problem. By contradiction, we consider the existence of a PL(7, 2) code and it is assumed that O = (0,…,0) is a codeword. Since O covers all words W ∈ β„€n which are distant two or less units from it, we focus our attention on the codewords which cover all words which are distant three units from O. Our idea is mostly based in cardinality restrictions on subsets of these codewords, being a natural adaptation of the one given by Horak in [12].

The next sections are organized as follows. In Section 2 some definitions, terminology and notation are given. Section 3 is devoted to the establishment of necessary conditions for the existence of PL(n, 2) codes for any positive integer n β‰₯ 7. Necessary conditions for the existence of PL(7, 2) codes are given in Section 4.

2 Definitions and notation

In this section we introduce some definitions and notation. The notation follows the one used by Horak [12].

Let (𝓒, ΞΌ) be a metric space, where 𝓒 is a nonempty set and ΞΌ a metric on 𝓒. Any subset π“œ of 𝓒 satisfying |π“œ| β‰₯ 2 is a code. The elements of 𝓒 are called words and, in particular, the elements of a code π“œ are called codewords.

A sphere centered at W ∈ 𝓒 with radius r, denoted by S(W, r), is defined as follows

S(W,r)={V∈S:ΞΌ(V,W)≀r}.

If W ∈ π“œ and V ∈ S(W, r), with V β‰  W, then we say that the codeword W covers the word V.

Definition 2.1

A code π“œ is a perfect r-error correcting code if:

  1. S(W, r) ∩ S(V, r) = βˆ… for any two distinct codewords W and V in π“œ;

  2. ⋃Wβˆˆπ“œS(W, r) = 𝓒.

In other words, π“œ is a perfect r-error correcting code if the spheres of radius r centered at codewords of π“œ form a partition of 𝓒. Equivalently, π“œ is a perfect r-error correcting code if the spheres of radius r centered at codewords of π“œ tile 𝓒.

When a code π“œ satisfies the condition i) in Definition 2.1, we say that π“œ is a r-error correcting code.

We are interested in dealing with metric spaces (β„€n, ΞΌL), where β„€n is the n-fold Cartesian product of the set of the integer numbers, with n a positive integer number, and ΞΌL is the Lee metric, that is, for any W, V ∈ β„€n, with W = (w1,…,wn) and V = (v1,…,vn), the Lee distance between W and V, shortly ΞΌL(W, V), is given by

ΞΌL(W,V)=βˆ‘i=1n|wiβˆ’vi|.

If π“œ βŠ‚ β„€n is a perfect r-error correcting code of (β„€n, ΞΌL), then π“œ is called a perfect r-error correcting Lee code of word length n over β„€, shortly a PL(n,r) code.

We detach the following necessary and sufficient condition on the Lee distance between two words to avoid superposition of spheres centered at them: Given W, V ∈ β„€n, with W β‰  V, and r a positive integer number, S(W, r) ∩ S(V, r) = βˆ… if and only if ΞΌL(W, V) β‰₯ 2r + 1.

Having in mind the Golomb-Welch conjecture, our aim is to give a contribution for the proof of the non-existence of PL(7, 2) codes. Our strategy is based on the assumption that their existence will bring strong cardinality restrictions on the cardinality of same codeword sets that we must identify and control.

Let us assume the existence of a PL(n, 2) code π“œ βŠ‚ β„€n, n β‰₯ 7, and suppose, without loss of generality, that O ∈ π“œ, with O = (0,…,0). Thus, all words W ∈ β„€n such that ΞΌL(W, O) ≀ 2 are covered by the codeword O. Taking into account Definition 2.1, for each word W ∈ β„€n satisfying ΞΌL(W, O) = 3 there exists a unique codeword V ∈ π“œ such that ΞΌL(W, V) ≀ 2. The conditions for the existence of PL(n, 2) codes derive essentially from the analysis of the codewords which cover all words W ∈ β„€n which are distant three units from O.

Let W ∈ β„€n such that ΞΌL(W, O) = 3. Then, W = (w1,…,wn) is of one and only one of the types:

  • β‹…

    [Β±3], if there exists i ∈ {1,…,n} so that |wi| = 3 and wj = 0 for all j ∈ {1,…,n} βˆ– {i};

  • β‹…

    [Β±2, Β±1], if |wi| = 2 and |wj| = 1 for some i, j ∈ {1,…,n}, and wk = 0 for all k ∈ {1,…,n} βˆ– {i, j};

  • β‹…

    [Β±13], if |wi| = |wj| = |wk| = 1 for some i, j, k ∈ {1,…,n}, and wl = 0 for all l ∈ {1,…,n} βˆ– {i, j, k}.

Let 𝓣 βŠ‚ π“œ be the set of the codewords which cover all the words W ∈ β„€n satisfying ΞΌL(W, O) = 3. Any codeword V ∈ 𝓣 is such that ΞΌL(V, O) = 5. In fact, since O and V are codewords in π“œ, to avoid superposition between them we must impose ΞΌL(V, O) β‰₯ 2 Γ— 2 + 1 = 5. On the other hand, if we suppose ΞΌL(V, O) β‰₯ 6, then for W such that ΞΌL(W, O) = 3 we get ΞΌL(V, W) β‰₯ 3.

Following the same idea used in the characterization of the words which are distant three units from O, we conclude that V ∈ 𝓣 is of one and only one of the types: [Β±5], [Β±4, Β±1], [Β±3, Β±2], [Β±3, Β±12], [Β±22, Β±1], [Β±2, Β±13] and [Β±15]. We will denote the subsets of 𝓣 containing codewords of each one of these types by, respectively, 𝓐, 𝓑, 𝓒, 𝓓, 𝓔, 𝓕 and 𝓖. Furthermore, we set a = |𝓐|, b = |𝓑|, c = |𝓒|, d = |𝓓|, e = |𝓔|, f = |𝓕| and g = |𝓖|, where |𝓐| denotes the cardinality of the set 𝓐 and so on.

Consider

I={+1,+2,…,+n,βˆ’1,βˆ’2,…,βˆ’n}

the set of signed coordinates. Let W, V ∈ β„€n, with W = (w1,…,wn) and V = (v1,…,vn). If iw|i| > 0 for i ∈ π“˜, then i and w|i| have the same sign. If iw|i| > 0 and iv|i| > 0, with i ∈ π“˜, then the |i|-th coordinates of W and V have the same sign and we say that W and V are sign equivalent in the |i|-th coordinate.

Let 𝓗 βŠ‚ β„€n. For i1, i2,…,ip ∈ π“˜, with p ≀ n and |i1|, |i2|,…,|ip| pairwise distinct, 𝓗i1i2…ip will denote the following set:

{W∈H:i1w|i1|>0∧i2w|i2|>0βˆ§β€¦βˆ§ipw|ip|>0}.

Given a positive integer number k and i ∈ π“˜, Hi(k) will denote:

{W∈H:iw|i|>0∧|w|i||=k}.

These sets are called index subsets of 𝓗. We note that, it makes no sense to consider 𝓗ij for i = j or i = βˆ’j, so, in the rest of the document, when we write 𝓗i1i2…ip, with 𝓗 βŠ‚ β„€n and i1, i2,…,ip ∈ π“˜, we assume |i1|, |i2|,…,|ip| pairwise distinct.

Consider W ∈ 𝓖. Since the codewords of 𝓖 are of type [Β±15], there are i, j, k, l, m ∈ π“˜ such that W ∈ 𝓖ijklm, where iw|i|, jw|j|, kw|k|, lw|l|, mw|m| > 0 and |w|i|| = |w|j|| = |w|k|| = |w|l|| = |w|m|| = 1. In this case i, j, k, l and m characterize the index distribution of W ∈ 𝓖. If we consider W ∈ 𝓕, since the codewords of 𝓕 are of type [Β±2, Β±13], there exist i, j, k, l ∈ π“˜ so that W ∈ 𝓕ijkl, more precisely, W∈Fi(2)∩Fj(1)∩Fk(1)∩Fl(1), where iw|i|, jw|j|, kw|k|, lw|l| > 0, |w|i|| = 2 and |w|j|| = |w|k|| = |w|l|| = 1, being characterized the index value distribution of W.

3 PL(n, 2) codes

In this section some necessary conditions for the existence of PL(n, 2) codes, for n β‰₯ 7, are given.

Let π“œ βŠ‚ β„€n be a PL(n, 2) code, with n β‰₯ 7. Suppose that O = (0,…,0) is a codeword of π“œ. Assume that 𝓣 βŠ‚ π“œ is the set of the codewords which cover all the words W ∈ β„€n satisfying ΞΌL(W, O) = 3. We have characterized in the previous section a partition of 𝓣 formed by the sets 𝓐, 𝓑, 𝓒, 𝓓, 𝓔, 𝓕 and 𝓖, composed, respectively, by codewords of types [Β±5], [Β±4, Β±1], [Β±3, Β±2], [Β±3, Β±12], [Β±22, Β±1], [Β±2, Β±13] and [Β±15].

We note that, the words of types:

  • β‹…

    [Β±3] must be covered by codewords of 𝓐 βˆͺ 𝓑 βˆͺ 𝓒 βˆͺ 𝓓;

  • β‹…

    [Β±2, Β±1] must be covered by codewords of 𝓑 βˆͺ 𝓒 βˆͺ 𝓓 βˆͺ 𝓔 βˆͺ 𝓕;

  • β‹…

    [Β±13] must be covered by codewords of 𝓓 βˆͺ 𝓔 βˆͺ 𝓕 βˆͺ 𝓖.

Let W ∈ β„€n such that W = (w1, …, wn) and ΞΌL(W, O) = 3. Suppose that W is a word of type [Β±2, Β±1]. Thus, there are i, j ∈ π“˜, with |i| β‰  |j|, such that, iw|i|, jw|j| > 0, |w|i|| = 2 and |w|j|| = 1. In these conditions we must impose, for instance, |Di(3)∩Dj(1)|≀1, otherwise, there are V, Vβ€²βˆˆDi(3)∩Dj(1), with V β‰  Vβ€², covering the same word W, contradicting the definition of PL(n, 2) code. In fact, supposing V∈Di(3)∩Dj(1)∩Dk(1) and Vβ€²βˆˆDi(3)∩Dj(1)∩Dl(1), we would have ΞΌL(V, W) = |v|i| – w|i|| + |v|k| – w|k|| = 2 and ΞΌL(Vβ€², W) = |v|i|β€²βˆ’w|i||+|v|l|β€²βˆ’w|l||=2. Having in view the word W similar conditions can be deduced to another sets of codewords, such as |(Di(3)∩Dj(1))βˆͺ(Ei(2)∩Ej)|≀1.

Taking into account the words of each one of the types [Β±3], [Β±2, Β±1] and [Β±13], and considering the sets of codewords that can cover them, we get the following lemmas.

Lemma 3.1

For each i ∈ π“˜, |AiβˆͺBi(4)βˆͺCi(3)βˆͺDi(3)|=1.

Proof

For each i ∈ π“˜ there exists a word W ∈ β„€n of type [Β±3], with W = (w1, …, wn), satisfying iw|i| > 0 and |w|i|| = 3. This word W must be covered by a codeword V ∈ 𝓐βˆͺ𝓑βˆͺ𝓒βˆͺ𝓓, in particular, V∈AiβˆͺBi(4)βˆͺCi(3)βˆͺDi(3). Thus, we conclude that |AiβˆͺBi(4)βˆͺCi(3)βˆͺDi(3)|β‰₯1. If, by contradiction, we assume |AiβˆͺBi(4)βˆͺCi(3)βˆͺDi(3)|β‰₯2, then there are two distinct codewords V and Vβ€² in AiβˆͺBi(4)βˆͺCi(3)βˆͺDi(3) satisfying ΞΌL(V, W) ≀ 2 and ΞΌL(Vβ€², W) ≀ 2, which contradicts the definition of PL(n, 2) code.  ░

Lemma 3.2

For each i, j ∈ π“˜, with |i| β‰  |j|,

|Bi(4)∩Bj(1)|+|Ci∩Cj|+|Di(3)∩Dj(1)|+|Ei(2)∩Ej|+|Fi(2)∩Fj(1)|=1.

Proof

For each i, j ∈ π“˜, with |i| β‰  |j|, there exists a word W ∈ β„€n of type [Β±2, Β±1], with W = (w1, …, wn), satisfying iw|i|, jw|j| > 0, |w|i|| = 2 and |w|j|| = 1. This word must be covered by a codeword V ∈ 𝓑βˆͺ𝓒βˆͺ𝓓βˆͺ𝓔βˆͺ𝓕, in particular, V∈(Bi(4)∩Bj(1))βˆͺ(Ci∩Cj)βˆͺ(Di(3)∩Dj(1))βˆͺ(Ei(2)∩Ej)βˆͺ(Fi(2)∩Fj(1)). Consequently, taking into account that 𝓑, 𝓒, 𝓓, 𝓔 and 𝓕 are disjoint sets,

|Bi(4)∩Bj(1)|+|Ci∩Cj|+|Di(3)∩Dj(1)|+|Ei(2)∩Ej|+|Fi(2)∩Fj(1)|β‰₯1.

If, by contradiction, we suppose

|Bi(4)∩Bj(1)|+|Ci∩Cj|+|Di(3)∩Dj(1)|+|Ei(2)∩Ej|+|Fi(2)∩Fj(1)|β‰₯2,

then, there are distinct codewords V and Vβ€² satisfying

V,Vβ€²βˆˆ(Bi(4)∩Bj(1))βˆͺ(Ci∩Cj)βˆͺ(Di(3)∩Dj(1))βˆͺ(Ei(2)∩Ej)βˆͺ(Fi(2)∩Fj(1)).

Consequently, ΞΌL(V, W) ≀ 2 and ΞΌL(Vβ€², W) ≀ 2, which contradicts the definition of perfect 2-error correcting code.  ░

Lemma 3.3

For each i, j, k ∈ π“˜, with |i|, |j| and |k| pairwise distinct,

|DijkβˆͺEijkβˆͺFijkβˆͺGijk|=1.

Proof

For each i, j, k ∈ π“˜, with |i|, |j| and |k| pairwise distinct, there is a word W ∈ β„€n of type [Β±13], with W = (w1, …, wn), such that, iw|i|, jw|j|, kw|k| > 0 and |w|i|| = |w|j|| = |w|k|| = 1. This word must be covered by a codeword V ∈ 𝓓ijk βˆͺ 𝓔ijk βˆͺ 𝓕ijk βˆͺ 𝓖ijk, therefore |𝓓ijk βˆͺ 𝓔ijk βˆͺ 𝓕ijk βˆͺ 𝓖ijk| β‰₯ 1. If, by contradiction, we suppose that |𝓓ijk βˆͺ 𝓔ijk βˆͺ 𝓕ijk βˆͺ 𝓖ijk| β‰₯ 2, then there are distinct codewords V, Vβ€² ∈ 𝓓ijk βˆͺ 𝓔ijk βˆͺ 𝓕ijk βˆͺ 𝓖ijk and, consequently, ΞΌL(V, W) ≀ 2 and ΞΌL(Vβ€², W) ≀ 2, contradicting the definition of PL(n, 2) code.  ░

Taking into account the number of words of each one of the types [Β±3], [Β±2, Β±1] and [Β±13], and considering the type of codewords which cover them, Horak has deduced in [12] the following proposition involving the parameters a = |𝓐|, b = |𝓑|, c = |𝓒|, d = |𝓓|, e = |𝓔|, f = |𝓕| and g = |𝓖|.

Proposition 3.4

The parameters a, b, c, d, e, f and g satisfy the system of equations

a+b+c+d=2nb+2c+2d+4e+3f=8n2d+e+4f+10g=8n3.

There exist many nonnegative integer solutions for this system of equations. However, we are interested in determining β€œgood” solutions, that is, solutions which do not contradict the definition of perfect 2-error correcting Lee code.

We may relate the cardinality of each set 𝓐, 𝓑, 𝓒, 𝓓, 𝓔, 𝓕 and 𝓖 with the cardinality of their index subsets. Taking into account, for instance, the set 𝓖, since the codewords of 𝓖 are of type [Β±15], we get

g=15βˆ‘i∈I|Gi|.

Besides, for i ∈ π“˜,

|Gi|=14βˆ‘j∈Iβˆ–{i,βˆ’i}|Gij|.

Analogous equalities for the other subsets of 𝓣 may be derived.

The analysis of the solutions for the system of equations presented in Proposition 3.4 will be focused essentially in the study of the cardinality of the index subsets of 𝓐, 𝓑, 𝓒, 𝓓, 𝓔, 𝓕 and 𝓖.

Looking at the words of type [Β±13], Horak proved in [12] the following proposition in which a relation between the cardinality of index subsets of 𝓓, 𝓔, 𝓕 and 𝓖 is given.

Proposition 3.5

For each i, j ∈ π“˜, |i| β‰  |j|,

|DijβˆͺEij|+2|Fij|+3|Gij|=2(nβˆ’2).

4 Conditions for the existence of PL(7, 2) codes

In this section we concentrate our attention on the search of necessary conditions for the existence of PL(7, 2) codes.

Let us suppose that π“œ βŠ‚ β„€7 is a PL(7, 2) code, with O = (0, …, 0) a codeword of π“œ. By Proposition 3.4, the parameters a, b, c, d, e, f and g satisfy:

a+b+c+d=14b+2c+2d+4e+3f=168d+e+4f+10g=280.

As we have said before, there are many nonnegative integer solutions for this system of equations, however we are only interested in those which do not contradict the definition of a perfect 2-error correcting Lee code. Since, g = |𝓖| is the variable with highest coefficient in the system and the codewords of 𝓖 are the ones which have more nonzero coordinates, a particular attention to the set 𝓖, more precisely, to the subsets 𝓖i, for i ∈ π“˜, will be given.

In [16], the following theorem which restricts the variation of |𝓖i|, for any i ∈ π“˜, was established.

Theorem 4.1

For each i ∈ π“˜, 3 ≀ |𝓖i| ≀ 8.

This theorem restricts the variation of g, in fact, since

g=15βˆ‘i∈I|Gi|,

taking into account that 3 ≀ |𝓖i| ≀ 8 for all i ∈ π“˜ and that |π“˜| = 14, we conclude that the solutions which do not contradict the definition of PL(7, 2) code must satisfy

9≀g≀22.

Our strategy to prove the non-existence of PL(7, 2) codes relies on restricting more and more the variation of |𝓖i|, for any i ∈ π“˜, more precisely, limiting more and more the variation of g.

In the following subsection we prove that |𝓖i| β‰  8 for all i ∈ π“˜.

4.1 Proof of |𝓖i| β‰  8 for any i ∈ π“˜

We will prove that |𝓖i| β‰  8 for any i ∈ π“˜ by contradiction. Let us suppose that there exists i ∈ π“˜ such that |𝓖i| = 8. Thus, since

|Gi|=14βˆ‘Ο‰βˆˆIβˆ–{i,βˆ’i}|GiΟ‰|,

we get

8=14βˆ‘Ο‰βˆˆIβˆ–{i,βˆ’i}|GiΟ‰|.

Consequently,

βˆ‘Ο‰βˆˆIβˆ–{i,βˆ’i}|GiΟ‰|=32.(1)

From Proposition 3.5 it follows that |𝓖iΟ‰| ≀ 3 for all Ο‰ ∈ π“˜ βˆ– {i, –i}. Particular attention will be given to the elements Ο‰ ∈ π“˜ βˆ– {i, –i} such that |𝓖iΟ‰| = 3 or |𝓖iΟ‰| = 2.

Throughout this subsection 𝓙 and π“š will denote the following sets:

J={j∈Iβˆ–{i,βˆ’i}:|Gij|=3}

and

K={k∈Iβˆ–{i,βˆ’i}:|Gik|=2}.

We begin by characterizing partially the index distribution of the codewords w1, …, W8 ∈ 𝓖i.

Proposition 4.2

If |𝓖i| = 8, i ∈ π“˜, then π“˜ βˆ– {i, –i} = 𝓙 βˆͺ π“š, with |𝓙| = 8 and |π“š| = 4. The partial index distribution of the codewords w1, …, W8 ∈ 𝓖i satisfies:

Table 1

where x, –x, y, –y ∈ 𝓙 and k1, …, k8 ∈ π“š. Consequently, for all W ∈ 𝓖i there exists a unique element k ∈ π“š such that W ∈ 𝓖ik.

Proof

Let i ∈ π“˜ such that |𝓖i| = 8. In these conditions, (1) is satisfied. By Proposition 3.5, for any Ο‰ ∈ π“˜ βˆ– {i, –i} we get |𝓖iΟ‰| ≀ 3. As |π“˜ βˆ– {i, –i}| = 12, taking into account (1) we conclude that there are, at least, eight elements Ο‰ ∈ π“˜ βˆ– {i, –i} satisfying |𝓖iΟ‰| = 3. We have just concluded that |𝓙| β‰₯ 8.

Let us consider

L={l∈Iβˆ–{i,βˆ’i}:|Gil|≀2}.

Observing that, 𝓙 βˆͺ 𝓛 = π“˜ βˆ– {i, –i}, 𝓙 ∩ 𝓛 = βˆ…, |π“˜ βˆ– {i, –i}| = 12 and |𝓙| β‰₯ 8, then |𝓛| ≀ 4. Thus, there are, at most, four distinct elements j ∈ 𝓙 such that –j ∈ 𝓛. Since |𝓙| β‰₯ 8, there exist x, y ∈ 𝓙, distinct, such that –x, –y ∈ 𝓙. Then, let us consider x, –x, y, –y ∈ 𝓙.

By definition of 𝓙, |𝓖ix| = |𝓖i,–x| = |𝓖iy| = |𝓖i, –y| = 3. Taking into account Lemma 3.3, the partial index distribution of the codewords w1, …, W8 ∈ 𝓖i must satisfy the conditions presented in the Table 2, in which W1 ∈ 𝓖ixy, W2 ∈ 𝓖i,x,–y and so on.

Table 2

Partial index distribution of the codewords of 𝓖i.

Looking at W1 ∈ 𝓖ixy, there are Ξ±, Ξ² ∈ π“˜ βˆ– {i, –i, x, –x, y, –y} such that W1 ∈ 𝓖ixyΞ±Ξ². Suppose that Ξ±,Ξ² ∈ 𝓙, that is, |𝓖iΞ±| = |𝓖iΞ²| = 3. Talking into account Lemma 3.3, |𝓖ixΞ±| = |𝓖iyΞ±| = |𝓖ixΞ²| = |𝓖iyΞ²| = 1. Besides, 𝓖ixΞ± = 𝓖iyΞ± = 𝓖ixΞ² = 𝓖iyΞ² = {W1}. Since |𝓖iΞ±| = 3, taking into account Table 2 and Lemma 3.3, 𝓖iΞ± βˆ– {W1} βŠ‚ {W5, W6, W8} and 𝓖iΞ² βˆ– {W1} βŠ‚ {W5, W6, W8}. As |𝓖iΞ± βˆ– {W1}| = |𝓖iΞ² βˆ– {W1}| = 2, there exists W ∈ {W5, W6, W8} such that W ∈ 𝓖iΞ±Ξ², which contradicts Lemma 3.3 since W, W1 ∈ 𝓖iΞ±Ξ². Therefore, there exists l1 ∈ 𝓛 so that W1 ∈ 𝓖ixyl1. Similarly, there are l2, l4, l5 ∈ 𝓛 such that W2 ∈ 𝓖i,x,–y,l2, W4 ∈ 𝓖i,–x,y,l4 and W5 ∈ 𝓖i,–x,–y,l5.

Let us consider W3 ∈ 𝓖ix. Having in view w1, W2 ∈ 𝓖ix and Lemma 3.3, there are Ξ±, Ξ², Ξ³ ∈ π“˜βˆ–{i,–i,x,–x,y,–y} so that W3 ∈ 𝓖ixΞ±Ξ²Ξ³. Assume that {Ξ±, Ξ², Ξ³} βŠ‚ 𝓙. Then, |𝓖iΞ±| = |𝓖iΞ²| = |!𝓖iΞ³| = 3. Accordingly, considering Lemma 3.3, we get |𝓖ixΞ±| = |𝓖ixΞ²| = |𝓖ixΞ³| = 1 and, as a consequence, 𝓖ixΞ± = 𝓖ixΞ² = 𝓖ixΞ³ = {W3}. Taking into account Table 2 and Lemma 3.3, we obtain: 𝓖iΞ± βˆ– {W3\\} βŠ‚ {W4,…,W8}; 𝓖iΞ² βˆ– {W3} βŠ‚ {W4,…,W8}; 𝓖iΞ³ βˆ– {W3} βŠ‚ {W4,…,W8}. Since |𝓖iΞ± βˆ– {W3}| = |𝓖iΞ² βˆ– {W3}| = |𝓖iΞ³ βˆ– {W3}| = 2 and |{W4,…,W8}| = 5, there exists W ∈ {W4,…,W8} such that W ∈ 𝓖iΡθ for Ξ΅, ΞΈ ∈ {Ξ±, Ξ², Ξ³}, which contradicts Lemma 3.3 since W, W3 ∈ 𝓖iΡθ. Thus, there exists l3 ∈ 𝓛 such that W3 ∈ 𝓖ixl3. Likewise, there are l6, l7, l8 ∈ 𝓛 such that W6 ∈ 𝓖i,–x,l6, W7 ∈ 𝓖iyl7 and W8 ∈ 𝓖i,–y,l8.

Therefore, for all W ∈ 𝓖i there exists l ∈ 𝓛 such that W ∈ 𝓖il.

By definition of 𝓛, |𝓖il| ≀ 2 for all l ∈ 𝓛. We have concluded before that |𝓛| ≀ 4. Since for any W ∈ 𝓖i there exists l ∈ 𝓛 such that W ∈ 𝓖il and |𝓖i| = 8, we must impose |𝓛| = 4 and |𝓖il| = 2 for any l ∈ 𝓛. That is, π“š = {k ∈ π“˜ {i, βˆ’i} : |𝓖ik| = 2} is such that |π“š| = 4. Consequently, for each W ∈ 𝓖i there exists a unique element k ∈ π“š such that W ∈ 𝓖ik. Furthermore, |𝓙| = 8, π“˜ {i, βˆ’i} = 𝓙 βˆͺ π“š and the partial index distribution of the codewords of 𝓖i satisfies the conditions which are given in the statement of this proposition. β–‘

The following result characterizes in more detail the set π“š and, consequently, the set 𝓙.

Proposition 4.3

If k ∈ π“š, then βˆ’k ∈ π“š.

Proof

We are assuming |𝓖i| = 8 for i ∈ π“˜. The partial index distribution of the codewords W 1, …,W8 ∈ 𝓖i satisfies the conditions enunciated in Proposition 4.2. We recall that, from this proposition it follows that π“˜ \{i, βˆ’i} = 𝓙 βˆͺ π“š, with |𝓙| = 8 and |π“š| = 4. Furthermore, {x, βˆ’x, y, βˆ’y} βŠ‚ 𝓙 and {k1, …, k8} = π“š.

Let us consider 𝓝 = 𝓙 {x, βˆ’x, y, βˆ’y} = {Ξ±, Ξ², Ξ³, Ξ΄}. We note that,

Iβˆ–{i,βˆ’i}={k1,…,k8}βˆͺ{x,βˆ’x,y,βˆ’y}βˆͺ{Ξ±,Ξ²,Ξ³,Ξ΄}.

By Proposition 4.2, for each W ∈ 𝓖i there exists a unique element k ∈ π“š such that W ∈ 𝓖ik. On the other hand, since |𝓖ij| = 3 for all j ∈ 𝓙, we have identified all codewords of 𝓖ix, 𝓖i, βˆ’x, 𝓖iy and 𝓖i, βˆ’y}. Thus, to characterize completely the index distribution of all codewords of 𝓖i we must fill in with elements of 𝓝 the empty entries of the table presented in Proposition 4.2.

Consider W1, W2, W3 ∈ 𝓖ix, see table in Proposition 4.2. Taking into account Lemma 3.3, the index distribution of the codewords of 𝓖ix must satisfy the conditions in Table 3.

Table 3

Partial index distribution of the codewords of 𝓖i.

Let us now consider the codeword W4 ∈ 𝓖i, k4, βˆ’x, y. Having in mind Lemma 3.3 we conclude that W4 βˆ‰ 𝓖α, otherwise we would get W1,W4 ∈ 𝓖iyΞ±. Suppose that W4 ∈ 𝓖β. In these conditions, W4, W2 ∈ 𝓖iΞ², with W4 ∈ 𝓖i,k4,βˆ’x,y,Ξ² and W2 ∈ 𝓖i,k2,x,βˆ’y,Ξ². Since |𝓖iΞ²| = 3 (Ξ² ∈ 𝓙), there exists W ∈ 𝓖i\{W1, W2, W3, W4} such that W ∈ 𝓖iΞ². By Table 3 we verify that W ∈ 𝓖i,Ξ²,βˆ’x βˆͺ 𝓖iΞ²y βˆͺ 𝓖i,Ξ²,βˆ’y. Consequently, taking into account W2 and W4, |𝓖iΞ²z| β‰₯ 2 for some z ∈ {βˆ’x, y, βˆ’y}, contradicting Lemma 3.3.

Therefore, W4 ∈ 𝓖γ βˆͺ 𝓖δ. By a similar reasoning, we are led to the conclusion that W5 ∈ 𝓖γ βˆͺ 𝓖δ.

We are assuming W3 ∈ 𝓖ik3xΞ³Ξ΄. As k3 ∈ π“š, by definition of π“š we get |𝓖ik3| = 2 . Thus, there exists k ∈ {k1, …, k8}\{k3} such that k = k3. We note that, k3 β‰  k1, k2, otherwise Lemma 3.3 is contradicted. Since W4, W5 ∈ 𝓖γ βˆͺ 𝓖δ, taking into account Lemma 3.3 we conclude that k3 β‰  k4, k5. Therefore, k ∈ {k6, k7, k8}. If k3 = k7, then Lemma 3.3 forces W7 ∈ 𝓖ik7yΞ±Ξ², which is a contradiction, since W1, W7 ∈ 𝓖iyΞ±. Then, k3 β‰  k7. By a similar reasoning we may conclude that k3 β‰  k8. Consequently, k3 = k6 and, applying once again Lemma 3.3, we must impose W6 ∈ 𝓖i,k3,βˆ’x,Ξ±,Ξ².

Note that |𝓖iΞ±| = |𝓖iΞ²| = 3. Since W4, W5 ∈ 𝓖γ βˆͺ 𝓖δ, we must obligate W7, W8 ∈ 𝓖α βˆͺ 𝓖β. Considering W1 and W2, Lemma 3.3 leads us to conclude that W7 ∈ 𝓖β and W8 ∈ 𝓖α.

Accordingly, the partial index distribution of the codewords of 𝓖i satisfies:

Table 4

Partial index distribution of the codewords of 𝓖i.

Note that, as |𝓖iΞ³| = |𝓖iΞ΄| = 3, the four empty entries of this table must be filled in with Ξ³ and Ξ΄. Thus, W4, W5, W7, W8 ∈ 𝓖γ βˆͺ 𝓖δ.

Consider the elements of π“š. By the analysis of the entries of the previous table, to avoid the contradiction of Lemma 3.3, one should have k1 = k5, k2 = k4 and k7 = k8. That is, π“š = {k1, k2, k3, k7} and the codewords of 𝓖i are characterize as it is presented in Table 5.

Table 5

Partial index distribution of the codewords of 𝓖i.

We intend to show that if k ∈ π“š, then βˆ’k ∈ π“š. Let us focus our attention on k3 ∈ π“š. We have concluded before that W3, W6 ∈ 𝓖ik3, with W3 ∈ 𝓖ik3xΞ³Ξ΄ and W6 ∈ 𝓖i,k3,βˆ’x,Ξ±,Ξ². In these conditions, βˆ’k3 ∈ π“˜ \({i,βˆ’i,x, βˆ’x,y,βˆ’y} βˆͺ 𝓝). That is, βˆ’k3 ∈ π“˜\({i,βˆ’i} βˆͺ 𝓙). Since π“˜ = {i, βˆ’i} βˆͺ 𝓙 βˆͺ π“š, then βˆ’k3 ∈ π“š.

Looking at the codewords W7, W8 ∈ 𝓖ik7, we get W7 ∈ 𝓖γ and W8 ∈ 𝓖δ, or, W7 ∈ 𝓖δ and W8 ∈ 𝓖γ. In both cases βˆ’k7 ∈ π“˜\({i, βˆ’i} βˆͺ 𝓙), accordingly βˆ’k7 ∈ π“š.

Now, π“š = {k1, k2, k3, k7} and βˆ’k3,-k7 ∈ π“š. Either k3 β‰ βˆ’k7 or k3 = βˆ’k7.

If k3 β‰  βˆ’k7, then βˆ’k ∈ π“š for all k ∈ π“š.

If k3 = βˆ’k7 and k1 = βˆ’k2, then βˆ’k ∈ π“š for all k ∈ π“š.

Assume that k3 = βˆ’k7 and k1 β‰  βˆ’k2. By this assumption it follows that βˆ’k1, βˆ’k2 ∈ 𝓝 = {Ξ±, Ξ², Ξ³, Ξ΄}. Thus, there are Ξ΅1, Ξ΅2 ∈ 𝓝 so that βˆ’k1 = Ξ΅1, βˆ’k2 = Ξ΅2 and the remaining elements of 𝓝, Ξ΅3 and Ξ΅4, satisfy Ξ΅3 = βˆ’Ξ΅4. As W1 ∈ 𝓖ik1xyΞ±, then βˆ’k1 ∈ {Ξ², Ξ³, Ξ΄}. On the other hand, since W2 ∈ 𝓖i,k2,x,βˆ’y,Ξ², then βˆ’k2 ∈ {Ξ±,Ξ³,Ξ΄}. We note that, as k1 β‰  k2, then βˆ’k1 β‰  βˆ’k2.

If βˆ’k1 = Ξ² and βˆ’k2 = Ξ±, then Ξ³ = βˆ’Ξ΄, which is a contradiction since W3 ∈ 𝓖ik3xΞ³Ξ΄.

If βˆ’k1 = Ξ² and βˆ’k2 = Ξ³, then Ξ± = βˆ’Ξ΄. Analyzing Table 5 and taking into account that W4 ∈ 𝓖γ βˆͺ 𝓖δ, we conclude that W4 ∈ 𝓖i,k2,βˆ’x,y,Ξ΄. Consequently, having in mind Lemma 3.3, W5 ∈ 𝓖i,k1,βˆ’x,βˆ’y,Ξ³, W7 ∈ 𝓖ik7yΞ²Ξ³ and W8 ∈ 𝓖i,k7,βˆ’y,Ξ±,Ξ΄, which is not possible since we are supposing Ξ± = βˆ’Ξ΄.

If βˆ’k1 = Ξ² and βˆ’k2 = Ξ΄, then Ξ± = βˆ’Ξ³. Consequently, W8 ∈ 𝓖i,k7,βˆ’y,Ξ±,Ξ΄, W7 ∈ 𝓖ik7yΞ²Ξ³ and W4 ∈ 𝓖i,k2,βˆ’x,y,Ξ΄. We get a contradiction since, by hypothesis, βˆ’k2 = Ξ΄.

Combining all possibilities for βˆ’k1 ∈ {Ξ², Ξ³, Ξ΄} and βˆ’k2 ∈ {Ξ±, Ξ³, Ξ΄}, by a similar reasoning we get always a contradiction. Therefore, βˆ’k ∈ π“š for all k ∈ π“š.  ░

From Proposition 4.2 we get π“˜\{i, βˆ’i} = 𝓙 βˆͺ π“š. We have just seen that, if k ∈ π“š then βˆ’k ∈ π“š. So, if j ∈ 𝓙 then βˆ’j ∈ 𝓙.

Until this moment we have focused our attention on the characterization of the codewords of 𝓖i. The two following propositions arise from the analysis of other type of codewords, in particular, codewords of 𝓓 βˆͺ 𝓔 βˆͺ 𝓕.

Proposition 4.4

If |𝓖i| = 8, i ∈ π“˜, then |𝓕i| = 0.

Proof

Let |𝓖i| = 8 for i ∈ π“˜. Suppose, by contradiction, that |𝓕i| > 0. Let U ∈ 𝓕i. Since the codewords of 𝓕 are of type [Β±2, Β±13], there exist $u1, u2, u3 ∈ π“˜ {i, βˆ’i}, with |u1|, |u2| and |u3| pairwise distinct, such that U ∈ 𝓕iu1u2u3.

By Proposition 4.2, π“˜\{i, βˆ’i} = 𝓙 βˆͺ π“š, therefore u1, u2, u3 ∈ 𝓙 βˆͺ π“š. Recall that |𝓖ij| = 3 for any j ∈ 𝓙. Then, by Proposition 3.5 one has |𝓕ij| = 0 for all j ∈ 𝓙. Consequently, u1, u2, u3 ∈ π“š. From Proposition 4.2 it follows that |π“š| = 4 and, taking into account Proposition 4.3, βˆ’k ∈ π“š for all k ∈ π“š. Thus, is not possible to have u1, u2, u3 ∈ π“š satisfying |u1|, |u2| and |u3| pairwise distinct, contradicting our assumption.β–‘

Proposition 4.5

For all j ∈ 𝓙, |𝓓ij βˆͺ 𝓔ij| = 1. For all k ∈ π“š, |𝓓ik βˆͺ 𝓔ik| = 4. Furthermore, if k ∈ π“š, the codewords U1, U2, U3, U4 ∈ 𝓓ik βˆͺ 𝓔ik are such that U1 ∈ 𝓓iku1 βˆͺ 𝓔iku1, U2 ∈ 𝓓iku2 βˆͺ 𝓔iku2, U3 ∈ 𝓓iku3 βˆͺ 𝓔iku3 and U4 ∈ 𝓓iku4 βˆͺ 𝓔iku4, with u1, u2 ∈ 𝓙, u1 β‰  u2, and u3, u4 ∈ π“š {k, βˆ’k}, with u3 = βˆ’u4.

Proof

From Proposition 3.5 we get

|DiΟ‰βˆͺEiΟ‰|+2|FiΟ‰|+3|GiΟ‰|=10(2)

for all Ο‰ ∈ π“˜\{i, βˆ’i}. By Proposition 4.4 we know that |𝓕i| = 0 and, consequently, |𝓕iΟ‰| = 0 for all Ο‰ ∈ π“˜\{i, βˆ’i}. As |𝓖ij| = 3 for any j ∈ 𝓙, from (2) we obtain |𝓓ij βˆͺ 𝓔ij| = 1 for all j ∈ 𝓙. Considering again (2), we conclude that |𝓓ik βˆͺ 𝓔ik| = 4 for each k ∈ π“š, since |𝓖ik| = 2 for all k ∈ π“š.

Let k ∈ π“š. Then, there exist V1, V2 ∈ 𝓖ik and U1, …, U4 ∈ 𝓓ik βˆͺ 𝓔ik. We note that, the codewords of 𝓓 are of type [Β±3,Β±12] and the codewords of 𝓔 are of type [Β±22, Β±1]. Thus, there are v1, …, v6, u1, …, u4 in π“˜ {i, βˆ’i, k, βˆ’k} such that:

Table 6

Index distribution of the codewords of 𝓖ik βˆͺ 𝓓ik βˆͺ 𝓔ik.

It should be pointed out that, by Lemma 3.3, v1, …, v6, u1, …, u4 must be pairwise distinct. Therefore, {v1, …, v6, u1, …, u4} = π“˜ {i, βˆ’i, k, βˆ’k}. By Proposition 4.2, π“˜ {i, βˆ’i} = 𝓙 βˆͺ π“š, with |𝓙| = 8 and |π“š| = 4. Furthermore, from Proposition 4.2, βˆ’k ∈ π“š. Then, {v1, …, v6, u1, …, u4} = 𝓙 βˆͺ π“š {k, βˆ’k}. Since V1,V2 ∈ 𝓖ik, with k ∈ π“š, taking into account Proposition 4.2 we must impose {v1, …, v6} βŠ‚ 𝓙. Consequently, without loss of generality, u1, u2 ∈ 𝓙 and u3, u4 ∈ π“š {k, βˆ’k}. Considering Proposition 4.2 we conclude that u3 = βˆ’u4. β–‘

We are now able to establish the main result of this paper.

Theorem 4.6

For any i ∈ π“˜, |𝓖i| β‰  8.

Proof

By contradiction, consider i ∈ π“˜ such that |𝓖i| = 8.

From Proposition 4.2 we have |π“š| = 4, so let k be an element of π“š. By Proposition 4.5, there exist U1, …, U4 ∈ 𝓓ik βˆͺ 𝓔ik whose index distribution satisfies the conditions presented in Table 7, where u, βˆ’u ∈ π“š\{k, βˆ’k} and j1, j2 ∈ 𝓙, with j1 β‰  j2. We note that, in these conditions, π“š = {k, βˆ’k, u, βˆ’u}.

Table 7

Index distribution of the codewords of 𝓓ik βˆͺ 𝓔ik.

Let us denote by 𝓗 the set of words of type [Β±2, Β±1]. Consider the words P1, P2 ∈ 𝓗 such that P1 ∈ Hi(2)∩Hj1(1)

and P2 ∈ Hi(2)∩Hj2(1) . The index distribution of the codewords of 𝓓ik βˆͺ 𝓔ik and the index value distribution of the words P1 and P2 are represented in the following table:

Table 8

Index distribution of U1, …, U4 ∈ 𝓓ik βˆͺ 𝓔ik and index value distribution of P1, P2 ∈ 𝓗i.

By definition of perfect 2-error correcting Lee code, for each P ∈ {P1, P2} there exists a unique codeword V ∈ 𝓣 such that ΞΌL(P, V) ≀ 2. Taking into account the type of words of 𝓗 as well as the fact of |𝓕i| = 0 (see Proposition 4.4), each word Pq Hi(2)∩Hjq, with jq ∈ π“˜\{i, βˆ’i}, is covered by a unique codeword

Vq∈(Bi(4)∩Bjq(1))βˆͺCijqβˆͺ(Di(3)∩Djq(1))βˆͺ(Ei(2)∩Ejq).(3)

Thus, we may consider U3 and U4 as possible codewords to cover P1 and P2, respectively.

Suppose that P1 is covered by U3 and P2 is covered by U4. Then, we must impose

U3∈(Di(3)∩Dk(1)∩Dj1(1))βˆͺ(Ei(2)∩Ek∩Ej1)

and

U4∈(Di(3)∩Dk(1)∩Dj2(1))βˆͺ(Ei(2)∩Ek∩Ej2),

which contradicts Lemma 3.2, since U3,U4 ∈ (Di(3)∩Dk(1))βˆͺ(Ei(2)∩Ek).

Therefore, either P1 is not covered by U3 or P2 is not covered by U4.

Without loss of generality, let us assume that P1 is not covered by U3. Note that, U3 ∈ 𝓓ikj1 βˆͺ 𝓔ikj1. As j1 ∈ 𝓙, by Proposition 4.5 we get |𝓓ij1 βˆͺ 𝓔ij1| = 1. Consequently, 𝓓ij1 βˆͺ 𝓔ij1 = {U3. Since we are assuming that U3 does not cover P1, considering (3), P1 is covered by a codeword V1 satisfying V1 ∈ (Bi(4)∩Bj1(1))βˆͺCij1.

Next, we will analyze, separately, the hypotheses:

  1. V1 ∈ Bi(4)∩Bj1(1);

  2. V1 ∈ 𝓒ij1.

  1. Assume that P1 is covered by V1 ∈ Bi(4)∩Bj1(1).

    Assuming that P1 is covered by V1 ∈ Bi(4)∩Bj1(1), by Lemma 3.1 we conclude |Bi(4)βˆ–{V1}βˆͺCi(3)βˆͺDi(3)|=0. Consequently, if U ∈ {U1,…,U4 is such that U ∈ 𝓓, then U∈Di(1). Furthermore, P2 must be covered by

    V2∈(Ci(2)∩Cj2(3))βˆͺ(Ei(2)∩Ej2).

    If V2 ∈ Ei(2) ∩ 𝓔j2, since j2 ∈ 𝓙 we conclude, by Proposition 4.5, that V2 = U4. Having in mind U1, U2 and U3, see Table 8, if U ∈ {U1, U2, U3} is such that U ∈ 𝓔, then U ∈ Ei(1), otherwise, U, U4 ∈ Ei(2) ∩ 𝓔k, contradicting Lemma 3.2. Therefore, since we have concluded before that {U1, U2, U3} ∩ Di(3) = βˆ…, we get U1, U2, U3 ∈ Di(1)βˆͺEi(1). Taking into account the index distribution of U1 and U2, we must have U1∈Du(3)orU2∈Dβˆ’u(3), otherwise we get U1, U2 ∈ (Di(1)∩Dk(3))βˆͺ(Ei(1)∩Ek(2)), contradicting, once again, Lemma 3.2.

    If V2∈Ci(2)∩Cj2(3), to avoid the contradiction of Lemma 3.2 we must impose U4∈Dk(3). Consequently, considering again Lemma 3.2, U1, U2, U3 ∈ Dk(1)βˆͺEk(1). We recall that, we have seen before that {U1, U2, U3} ∩ Di(3)=βˆ…. Thus, in these conditions, U1∈Du(3)orU2∈Dβˆ’u(3), otherwise, U1, U2 ∈ Ei(2)∩Ek(1), contradicting again Lemma 3.2.

    Therefore, in both cases, supposing V2 ∈ Ei(2)∩Ej2orV2∈Ci(2)∩Cj2(3), we conclude that U1∈Du(3) or U2∈Dβˆ’u(3).

    Suppose, without loss of generality, that U1∈Du(3). As u ∈ π“š, by Proposition 4.5 there are U5, U6 ∈ 𝓓iuβˆͺ𝓔iu satisfying U5 ∈ 𝓓iuj3 βˆͺ 𝓔iuj3 and U6 ∈ 𝓓iuj4 βˆͺ 𝓔iuj4, with j3, j4 ∈ 𝓙 distinct. Note that, j1,…,j4 ∈ 𝓙 are pairwise distinct, since by Proposition 4.5 we have |𝓓ij βˆͺ 𝓔ij| = 1 for all j ∈ 𝓙.

    Let us consider P3∈Hi(2)∩Hj3(1)andP4∈Hi(2)∩Hj4(1). Table 9 summarizes the conditions that the index distribution, and, in some cases, the index value distribution, of the codewords and words described until now, must satisfy.

    Table 9

    Index conditions on 𝓑i βˆͺ 𝓓i βˆͺ 𝓔i and on 4 words of type [Β±2, Β±1].

    Taking into account the words P3 and P4 we may conclude, as we have concluded before for P1 and P2, that either P3 is not covered by U5 or P4 is not covered by U6. In fact, if U5 covers P3 and U6 covers P4, then U5, U6 ∈ (Di(3)∩Du(1))βˆͺ(Ei(2)∩Eu), contradicting Lemma 3.2. Let us assume, without loss of generality, that P3 is not covered by U5. By Proposition 4.5 it follows that |𝓓ij3 βˆͺ 𝓔ij3| = 1. Consequently, 𝓓ij3 βˆͺ 𝓔ij3 = {U5}. As a consequence of the assumption V1 ∈ Bi(4)∩Bj1(1) we get |Bi(4)βˆ–{V1}βˆͺCi(3)βˆͺDi(3)|=0. Thus, under these conditions and taking into account (3), P3 must be covered by a codeword V3 satisfying V3 ∈ Ci(2)∩Cj3(3). Consequently, U5∈Du(3), otherwise, U5 ∈ (Di(1)∩Dj3(3))βˆͺ(Ei(2)∩Ej3)βˆͺ(Ei∩Ej3(2)) and contradicts with the codeword V3 Lemma 3.2. However, U1, U5∈Du(3), contradicting Lemma 3.1.

    Accordingly, P1 can not be covered by the codeword V1 ∈ Bi(4)∩Bj1(1).

  2. Assume that P1 is covered by V1 ∈ 𝓒ij1.

    Since V1 ∈ 𝓒, then V1 is a codeword of type [Β±3, Β±2]. According with what is being supposed, V1 ∈ Ci(3)∩Cj1(2)orV1∈Ci(2)∩Cj1(3). Consider U3 ∈ 𝓓ikj1 βˆͺ 𝓔ikj1. In order to have Lemma 3.2 fulfilled we must force U3 ∈ Di(1)∩Dk(3)∩Dj1(1). Schematically, we get Table 10.

    Table 10

    Index distribution on 𝓒i βˆͺ 𝓓i βˆͺ 𝓔i and on 2 words of type [Β±2, Β±1].

    Taking into account U3, by Lemma 3.2 we must have U1, U2, U4 ∈ Dk(1)βˆͺEk(1). Besides, U1∈Du(3)orU2∈Dβˆ’u(3), otherwise, U1, U2 ∈ (Di(3)∩Dk(1))βˆͺ(Ei(2)∩Ek(1)), contradicting Lemma 3.2.

    Let us assume, without loss of generality, that U1∈Du(3),

    Proceeding as in the previous case, we will consider U5 ∈ 𝓓iuj3 βˆͺ 𝓔iuj3 and U6 ∈ 𝓓iuj4 βˆͺ 𝓔iuj4, with j3, j4 ∈ 𝓙 and distinct. We will consider also P3 ∈ Hi(2)∩Hj3(1) and P4 ∈ Hi(2)∩Hj4(1). Gathering the information obtained so far, one has the index distribution presented in Table 11.

    Table 11

    Index distribution on 𝓒i βˆͺ 𝓓i βˆͺ 𝓔i and on 4 words of type [Β±2, Β±1].

    As seen in the previous case, either U5 does not cover P3 or U6 does not cover P4. Assume, without loss of generality, that P3 is not covered by U5. By Proposition 4.5 we get 𝓓ij3 βˆͺ 𝓔ij3 = {U5}. Therefore, considering (3), P3 must be covered by a codeword V3 ∈ (Bi(4)∩Bj3(1)) βˆͺ 𝓒ij3. If V3 ∈ 𝓒ij3, then, by Lemma 3.2, we must impose U5 ∈ Du(3) and, consequently, | Du(3)| β‰₯ 2, contradicting Lemma 3.1. Accordingly, V3 ∈ (Bi(4)∩Bj3(1)).

    Taking into account Lemma 3.1, |Bi(4)βˆ–{V3}βˆͺCi(3)βˆͺDi(3)|=0. Thus, by (3) we may conclude that P4 must be covered by a codeword

    V4∈(Ci(2)∩Cj4(3))βˆͺ(Ei(2)∩Ej4).

    Note that, if V4∈Ci(2)∩Cj4(3), then, by Lemma 3.2, U6 ∈ Du(3) implying | Du(3)| β‰₯ 2 and contradicting Lemma 3.1. Thus, V4 ∈ Ei(2)∩Ej4. By Proposition 4.5, |𝓓ij4 βˆͺ 𝓔ij4| = 1 leading to 𝓓ij4 βˆͺ 𝓔ij4 = {U6 and, consequently, V4 = U6. Since U1 ∈ Di(1)∩Dk(1)∩Du(3), taking into account Lemma 3.2, we must force U6 ∈ Ei(2)∩Eu(1)∩Ej4(2). The index distribution, and, in some cases the index value distribution, of the codewords and words which we are dealing with are presented in Table 12.

    Table 12

    Index distribution on 𝓑i βˆͺ 𝓒i βˆͺ 𝓓i βˆͺ 𝓔i and on 4 words of type [Β±2, Β±1].

    Let us now focus our attention on –u ∈ π“š. By Proposition 4.5, there are codewords U7, U8 ∈ 𝓓i,–u βˆͺ 𝓔i,–u, so that, U7 ∈ 𝓓i,–u,j5 βˆͺ 𝓔i,–u,j5 and U8 ∈ 𝓓i,–u,j6 βˆͺ 𝓔i,–u,j6, with j5, j6 ∈ 𝓙 distinct. Note that, by Proposition 4.5, |𝓓ij βˆͺ 𝓔ij| = 1 for all j ∈ 𝓙, and so j1,…,j6 are pairwise distinct. Taking into account the existence of the words P5 ∈ Hi(2)∩Hj5(1) and P6 ∈ Hi(2)∩Hj6(1), we obtain the index distribution presented schematically in Table 13.

    Table 13

    Index distribution on 𝓑i βˆͺ 𝓒i βˆͺ 𝓓i βˆͺ 𝓔i and on 6 words of type [Β±2, Β±1].

    By a similar reasoning to the one done with the words P1, P2, P3, P4 ∈ Hi(2), we conclude that either P5 is not covered by U7 or P6 is not covered by U8. Let us assume, without loss of generality, that U7 does not cover P5. Then, considering (3) we are lead to conclude that P5 must be covered by a codeword

    P5∈(Bi(4)∩Bj5(1))βˆͺ(Cij5).

    As V3 ∈ Bi(4)∩Bj3(1), by Lemma 3.1, P5 ∈ Ci(2)∩Cj5(3). Consequently, taking into account Lemma 3.2, we must force U7∈Dβˆ’u(3).

    Focus our attention on the codeword U2 ∈ 𝓓i,k,–u βˆͺ 𝓔i,k,–u. Having in mind the index value distribution of the codewords V3, U3 and U7 and considering Lemma 3.1, we conclude that U2 ∈ 𝓔i. Consequently, either U2 ∈ 𝓔i ∩ Ek(2) or U2 ∈ 𝓔i ∩ Eβˆ’u(2). If U2 ∈ 𝓔i ∩ Ek(2), then the index value distribution of U2 and U3 contradicts Lemma 3.2. If U2 ∈ 𝓔i ∩ Eβˆ’u(2), the index value distribution of U2 and U7 contradicts also Lemma 3.2.

    In both hypotheses, P1 covered by V1 ∈ Bi(4)∩Bj1(1) or P1 covered by V1 ∈ 𝓒ij1, we get a contradiction.  ░

We have proved in [16] that for each i ∈ π“˜, 3 ≀ |𝓖i| ≀ 8. From last theorem it follows immediately:

Corollary 4.7

For any i ∈ π“˜, 3 ≀ |𝓖i| ≀ 7.

Since g = |𝓖| = 15βˆ‘i∈I|Gi|, the required solutions for the system of equations presented in Proposition 3.4 must satisfy 9 ≀ g ≀ 19. As we have said before, our strategy to prove the non-existence of PL(7, 2) codes consists in getting a minimum range for the variation of |𝓖i|, with i ∈ π“˜, and consequently to reduce the number of solutions for the referred system of equations.

We have already started working on the analysis of other values for |𝓖i|, with i ∈ π“˜, which brings increased difficulties, imposing new strategies and techniques. It seems that our intuition on the new strategy to be applied (from now on) for the proving of the non-existence of PL(7, 2) codes will be successful.

Acknowledgement

This work was partially supported by Portuguese funds through CIDMA (Center for Research and Development in Mathematics and Applications) and FCT (Foundation for Science and Technology) within project UID/MAT/04106/2013.

References

  • [1]

    Golomb S.W., Welch L.R., Perfect codes in the Lee metric and the packing of polyominoes, SIAM J. Appl. Math., 1970, 18, 302-317.Β CrossrefGoogleΒ Scholar

  • [2]

    Lee C.Y., Some properties of nonbinary error-correcting codes, IRE Trans. Inf. Theory, 1958, 4, 72-82.Β GoogleΒ Scholar

  • [3]

    Golomb S.W., Welch L.R., Algebraic coding and the Lee metric, In: Error Correcting Codes, Wiley, New York, 1968, 175-189.Β GoogleΒ Scholar

  • [4]

    Barg A., Mazumdar A., Codes in permutations and error correction for rank modulation, IEEE Trans. Inf. Theory, 2010, 56(7), 3158-3165.Β WebΒ ofΒ ScienceCrossrefGoogleΒ Scholar

  • [5]

    Blaum M., Bruck J., Vardy A., Interleaving schemes for multidimensional cluster errors, IEEE Trans. Inf. Theory, 1998, 44, 730-743.Β CrossrefGoogleΒ Scholar

  • [6]

    Etzion T., Yaakobi E., Error-correction of multidimensional bursts, IEEE Trans. Inf. Theory, 2009, 55, 961-976.Β WebΒ ofΒ ScienceCrossrefGoogleΒ Scholar

  • [7]

    Roth R.M., Siegel P.H., Lee-metric BCH codes and their application to constrained and partial-response channels, IEEE Trans. Inf. Theory, 1994, 40, 1083-1096.Β CrossrefGoogleΒ Scholar

  • [8]

    Stein S., SzabΓ³ S., Algebra and Tiling: Homomorphisms in the Service of Geometry, In: Carus Mathematical Monographs, Vol. 25, Mathematical Association of America, 1994.Β GoogleΒ Scholar

  • [9]

    Gravier S., Mollard M., Payan CH., On the nonexistence of three-dimensional tiling in the Lee metric, European J. Combinatorics, 1998, 19, 567-572.Β CrossrefGoogleΒ Scholar

  • [10]

    Horak P., Tilings in Lee metric, European J. Combinatorics, 2009, 30, 480-489.Β WebΒ ofΒ ScienceCrossrefGoogleΒ Scholar

  • [11]

    Ε pacapan S., Non-existence of face-to-face four dimensional tiling in the Lee metric, European J. Combinatorics, 2007, 28, 127-133.Β CrossrefGoogleΒ Scholar

  • [12]

    Horak P., On perfect Lee codes, Discrete Mathematics, 2009, 309, 5551-5561.Β CrossrefWebΒ ofΒ ScienceGoogleΒ Scholar

  • [13]

    Horak P., Grosek O., A new approach towards the Golomb-Welch conjecture, European J. Combinatorics, 2014, 38, 12-22.Β CrossrefWebΒ ofΒ ScienceGoogleΒ Scholar

  • [14]

    Post K. A., Nonexistence theorem on perfect Lee codes over large alphabets, Inf. Control, 1975, 29, 369-380.Β CrossrefGoogleΒ Scholar

  • [15]

    Ε pacapan S., Optimal Lee-type local structures in Cartesian products of cycles and paths, SIAM J. Discrete Mathematics, 2007, 21, 750-762.Β CrossrefWebΒ ofΒ ScienceGoogleΒ Scholar

  • [16]

    Cruz C.N., D’azevedo Breda A. M., Some insights about PL(7, 2) codes, In: ATINER’S Conference Paper Series, No: MAT2013-0476 (7th Annual International Conference on Mathematics, June 2013, Athens, Greece), Athens Institute for Education and Research, 2013.Β GoogleΒ Scholar

About the article

Received: 2016-08-26

Accepted: 2018-02-09

Published Online: 2018-04-02


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 311–325, ISSN (Online) 2391-5455, DOI:Β https://doi.org/10.1515/math-2018-0027.

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© 2018 Cruz and ďAzevedo Breda, published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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