Perfect codes in the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si27.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mi>l</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:math>metric

Campello, Antonio; Jorge, Grasiele C.; Strapasson, João E.; Costa, Sueli I. R.

doi:10.1016/j.ejc.2015.11.002

Cited by 24 publications

(37 citation statements)

References 14 publications

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“…An example where the inequality of Theorem 4 is strict can be given by considering the chain Z 2 6 ⊇ C 1 ⊇ C 2 , where C 1 = (1, 2) and C 2 = (2, 4) along with the parameters r 1 = 1, r 2 = 2 and h 1 = (4, 1), h 2 = (3, 0) ∈ Z 2 6 . In this example, it can be shown that min{6 2 , 3d 1 Lee , d 2 Lee } = 4 < 5 = d 1 min (Λ D ) < 6 2 . We remark that when the vectors h 1 , ..., h r a used in Construction D satisfy the hypotheses of Theorem 2, we can obtain the density of Λ D relative to l 1 -distance from Theorems 2 and 4.…”

Section: Connections Between Constructions D D and Dmentioning

confidence: 89%

“…We remark that Example 3 does not contradict the conjecture regarding Corollary 4. Indeed, we have (3,6), (6,3), (1,2), (4,8), (7,5), (2, 1), (5,7), (8,4)}.…”

Section: Connections Between Constructions D D and Dmentioning

confidence: 99%

“…There are several constructions involving linear codes and lattices, for example the Construction A [3] and the Constructions D, D and D [18]. These constructions appear in recent works such as [1], [4], [5], [6], [7], [12], [13], [14] and [20]. Given a lattice, a problem of interest is to determine its minimum distance in a given metric.…”

Section: Introductionmentioning

confidence: 99%

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Bounds for the $$l_1$$ l 1 -distance of q-ary lattices obtained via Constructions D, D $$^{'}$$ ′ and $$\overline{\text{ D }}$$ D ¯

Strey

Costa

2017

Comp. Appl. Math.

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Lattices have been used in several problems in coding theory and cryptography. In this paper we approach q-ary lattices obtained via Constructions D, D and D. It is shown connections between Constructions D and D . Bounds for the minimum l 1 -distance of lattices Λ D , Λ D and Λ D and, under certain conditions, a generator matrix for Λ D are presented. In addition, when the chain of codes used is closed under the zero-one addition, we derive explicit expressions for the minimum l 1 -distances of the lattices Λ D and Λ D attached to the distances of the codes used in these constructions.

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Section: Connections Between Constructions D D and Dmentioning

confidence: 89%

“…We remark that Example 3 does not contradict the conjecture regarding Corollary 4. Indeed, we have (3,6), (6,3), (1,2), (4,8), (7,5), (2, 1), (5,7), (8,4)}.…”

Section: Connections Between Constructions D D and Dmentioning

confidence: 99%

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Bounds for the $$l_1$$ l 1 -distance of q-ary lattices obtained via Constructions D, D $$^{'}$$ ′ and $$\overline{\text{ D }}$$ D ¯

Strey

Costa

2017

Comp. Appl. Math.

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“…Let p(n, e) = e i=1 2 i e−i+1 j=1 j 2 e−j i−1 n−1 i−1 . Rearranging the sum and using that a b = 0 for b > a we obtain the expression p(n, e) = e i=0 2i 2 k(n − 1, e − i) (2) for every n, e ≥ 1. In other words p(n, e) is the coefficient of x e of the convolution of the generating functions f (x) = ∞ i=0 2i 2 x i and g(x) = ∞ i=0 k(n − 1, i)x i .…”

Section: Introductionmentioning

confidence: 99%

On the non-existence of linear perfect Lee codes: The Zhang–Ge condition and a new polynomial criterion

Qureshi

2020

European Journal of Combinatorics

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The Golomb-Welch conjecture (1968) states that there are no e-perfect Lee codes in Z n for n ≥ 3 and e ≥ 2. This conjecture remains open even for linear codes. A recent result of Zhang and Ge establishes the non-existence of linear e-perfect Lee codes in Z n for infinitely many dimensions n, for e = 3 and 4. In this paper we extend this result in two ways. First, using the non-existence criterion of Zhang and Ge together with a generalized version of Lucas' theorem we extend the above result for almost all e (i.e. a subset of positive integers with density 1). Namely, if e contains a digit 1 in its base-3 representation which is not in the unit place (e.g. e = 3, 4) there are no linear e-perfect Lee codes in Z n for infinitely many dimensions n. Next, based on a family of polynomials (the Q-polynomials), we present a new criterion for the non-existence of certain lattice tilings. This criterion depends on a prime p and a tile B. For p = 3 and B being a Lee ball we recover the criterion of Zhang and Ge.The Lee metric was introduced for transmission of signals over noisy channels in [13] for codes with alphabet Z p with p a prime number, then it was extended to alphabets Z q (q ∈ Z + ) and Z in [5], [6]. One of the most central question on codes in the Lee metric is regarding the 2 existence of such codes. In [6], Golomb and Welch showed that there are e-perfect Lee codes C ⊆ Z 2 for each value of e ≥ 1, and there are 1-perfect Lee codes C ⊆ Z n for each value of n ≥ 2. They also proved that for fixed dimension n ≥ 3, there is a radius e n > 0 (e n unspecified) such that there are no e-perfect Lee codes C ⊆ Z n for e ≥ e n and conjectured that it is possible to take e n = 2 (i.e. no e-perfect Lee codes in Z n exist for n ≥ 3 and e ≥ 2). This conjecture has been the main motive power behind the research in the area. A recent survey of papers on the Golomb-Welch conjecture is provided in [10]. Next we mention some of them. Explicit bounds for e n for periodic perfect Lee codes were obtained by K. A. Post [17], namely e n = n − 1 for 3 ≤ n ≤ 5 and e n = √ 2 2 n − 1 4 (3 √ 2 − 2) for n ≥ 6; and by P. Lepistö [14] who proved that an e-perfect Lee code must satisfy n ≥ (e + 2) 2 /2.1 if e ≥ 285. Recently, P. Horak and D. Kim [10] proved that the above results hold in general, that is, without the restriction of periodicity. The Golomb-Welch conjecture was also proved for dimensions 3 ≤ n ≤ 5 and radii e ≥ 2 [4], [21], [7] and for (n, e) = (6, 2) [8]. Recently, several papers have focus on the study of the Golomb-Welch conjecture for fixed radius e and large dimensions n. The case e = 2 has been treated in [11], [12] and [19]. In the linear case, it is proved that LPL(n, 2) = ∅ for infinitely many dimensions n. A new criterion for the non-existence of perfect Lee codes was presented by T. Zhang and G. Ge in [22] and it was used for the authors to obtain non-existence results for radii e = 3 and e = 4. This criterion states that if a pair (n, e) of positive integers verifies the congruence system (3) and k(n, e) is squar...

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“…Caso contrário, podemos apenas garantir a existência de um conjunto minimal de geradores. Por exemplo, o código linear C = (2, 4) = {(0, 0), (2,4), (4, 2)} ⊆ Z 2 6 não possui base, uma vez que todo subconjunto não vazio de Cé linearmente dependente. Para cada par de vetores x = (x 1 , ..., x n ) e y = (y 1 , ..., y n ) em Z n q , o produto interno de x e yé definido como x, y = x 1 y 1 + • • • + x n y n .…”

Section: Introductionunclassified

Limitantes para a distância da soma em reticulados obtidos via Construção D e suas variações

Strey¹,

Costa²

2016

Anais De XXXIV Simpósio Brasileiro De Telecomunicações

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Resumo-Reticulados vem sendo utilizados na abordagem de vários problemas em Teoria de Códigos e Criptografia. Neste trabalho, abordamos as Construções D, D and D, fornecemos cotas inferiores e superiores para a distância mínima em relaçãò a métrica da soma do reticulado Λ D em função das distâncias dos códigos lineares utilizados em sua construção. Também apresentamos, quando a cadeia de códigos usadaé fechada sob a adição zero-um, expressões para as distâncias mínimas em relaçãoà métrica da soma dos reticulados Λ D e ΛD em função das distâncias dos códigos utilizados em suas respectivas construções.

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Perfect codes in thelpmetric

Cited by 24 publications

References 14 publications

Bounds for the $$l_1$$ l 1 -distance of q-ary lattices obtained via Constructions D, D $$^{'}$$ ′ and $$\overline{\text{ D }}$$ D ¯

Bounds for the $$l_1$$ l 1 -distance of q-ary lattices obtained via Constructions D, D $$^{'}$$ ′ and $$\overline{\text{ D }}$$ D ¯

On the non-existence of linear perfect Lee codes: The Zhang–Ge condition and a new polynomial criterion

Limitantes para a distância da soma em reticulados obtidos via Construção D e suas variações

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