On the use of Lee-codes for constructing multiple-valued error-correcting decision diagrams

Astola, Helena; Stanković, Stanislav

doi:10.1109/isccsp.2012.6217864

Cited by 5 publications

(3 citation statements)

References 14 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Camarero

Martínez

2016

IEEE Trans. Inform. Theory

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A construction of 2-quasi-perfect Lee codes is given over the space Z n p for p prime, p ≡ ±5 (mod 12) and n = 2[ p 4 ]. It is known that there are infinitely many such primes. Golomb and Welch conjectured that perfect codes for the Lee-metric do not exist for dimension n ≥ 3 and radius r ≥ 2. This conjecture was proved to be true for large radii as well as for low dimensions. The codes found are very close to be perfect, which exhibits the hardness of the conjecture. A series of computations show that related graphs are Ramanujan, which could provide further connections between Coding and Graph Theories.

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Section: Introductionmentioning

confidence: 99%

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

Camarero

Martínez

2016

IEEE Trans. Inform. Theory

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“…In data transmission, the Lee metric can be used in phase modulation schemes, since corrupted digits of phasemodulated signals are more likely to have only slightly different phase than greatly different phase compared to the original signal [8]. There have been some more recent applications of Lee-codes to, for example, VLSI decoders and fault-tolerant logic, which are discussed in [9], [10] and [11].…”

Section: Introductionmentioning

confidence: 99%

On the linear programming bound for linear Lee codes

Astola

Tăbuş

2016

SpringerPlus

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Based on an invariance-type property of the Lee-compositions of a linear Lee code, additional equality constraints can be introduced to the linear programming problem of linear Lee codes. In this paper, we formulate this property in terms of an action of the multiplicative group of the field on the set of Lee-compositions. We show some useful properties of certain sums of Lee-numbers, which are the eigenvalues of the Lee association scheme, appearing in the linear programming problem of linear Lee codes. Using the additional equality constraints, we formulate the linear programming problem of linear Lee codes in a very compact form, leading to a fast execution, which allows to efficiently compute the bounds for large parameter values of the linear codes.

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“…Using space embeddings, Jiang et al gave a method in [15] to construct Charge-Constrained Rank-Modulation codes (CCRM codes) from Lee error-correcting codes, which could be employed for flash memories. Astola and Stankovic considered in [7] Lee codes to build decision diagrams.…”

Section: Introductionmentioning

confidence: 99%

Quasi-Perfect Lee Codes from Quadratic Curves over Finite Fields

Mesnager,

Tang,

2016

Preprint

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Golomb and Welch conjectured in 1970 that there only exist perfect Lee codes for radius t = 1 or dimension n = 1, 2. It is admitted that the existence and the construction of quasi-perfect Lee codes have to be studied since they are the best alternative to the perfect codes. In this paper we firstly highlight the relationships between subset sums, Cayley graphs, and Lee linear codes and present some results. Next, we present a new constructive method for constructing quasi-perfect Lee codes. Our approach uses subsets derived from some quadratic curves over finite fields (in odd characteristic) to derive two classes of 2quasi-perfect Lee codes are given over the space Z n p for n = p k +1 2 (with p ≡ 1, −5 mod 12 and k is any integer, or p ≡ −1, 5 mod 12 and k is an even integer) and n = p k −1 2 (with p ≡ −1, 5 mod 12, k is an odd integer and p k > 12), where p is an odd prime. Our codes encompass the quasi-perfect Lee codes constructed recently by Camarero and Martínez. Furthermore, we solve a conjecture proposed by Camarero and Martínez (in "quasi-perfect Lee codes of radius 2 and arbitrarily large dimension", IEEE Trans. Inf. Theory, vol. 62, no. 3, 2016) by proving that the related Cayley graphs are Ramanujan or almost Ramanujan. The Lee codes presented in this paper have applications to constrained and partial-response channels, in flash memories and decision diagrams.

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On the use of Lee-codes for constructing multiple-valued error-correcting decision diagrams

Cited by 5 publications

References 14 publications

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

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

On the linear programming bound for linear Lee codes

Quasi-Perfect Lee Codes from Quadratic Curves over Finite Fields

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