New upper bounds on Lee codes

Quistorff, Jörn

doi:10.1016/j.dam.2006.01.009

Cited by 10 publications

(6 citation statements)

References 13 publications

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“…However, for these distance measures other upper bounds are known. We recommend van Lint (1992) and Roth (2006) for the Hamming metric and Quistorff (2006) for the Lee metric. Note that it is enough for d to be a metric on Q for Corollary 4.1 to hold, there is no need for d to be a metric on the whole interval (−π , π ].…”

Section: Resultsmentioning

confidence: 99%

Generalized upper bounds on the minimum distance of PSK block codes

Laksman

Lennerstad

Nilsson

2014

IMA Journal of Mathematical Control and Information

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This paper generalizes previous optimal upper bounds on the minimum Euclidean distance for phase shift keying (PSK) block codes, that are explicit in three parameters: alphabet size, block length and code size. The bounds are primarily generalized from codes over symmetric PSK to codes over asymmetric PSK and also to general alphabet size. Furthermore, block codes are optimized in the presence of other types of noise than Gaussian, which induces also non-Euclidean distance measures. In some instances, codes over asymmetric PSK prove to give higher Euclidean distance than any code over symmetric PSK with the same parameters. We also provide certain classes of codes that are optimal among codes over symmetric PSK.

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Section: Resultsmentioning

confidence: 99%

Generalized upper bounds on the minimum distance of PSK block codes

Laksman

Lennerstad

Nilsson

2014

IMA Journal of Mathematical Control and Information

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“…Because of this property, the Lee distance is used in certain communication systems for information transmission (so called 'phase modulated systems', see [7,Chapter 8]). Generally, it is an interesting and nontrivial problem to determine A L q (n, d) for given q, n, d. Quistorff made a table of upper bounds on A L q (n, d) based on analytic arguments [20]. H. Astola and I. Tabus calculated several new upper bounds by linear programming [2], using an adaptation of the classical Delsarte bound based on pairs of codewords [9] (see also [1]).…”

Section: Introductionmentioning

confidence: 99%

“…//Start with |D| = 1. foreach n = (n 1 , n 2 ) ∈ N foreach λ = (λ 1 , λ 2 ) n with height(λ 1 ) ≤ q/2 + 1, height(λ 2 ) ≤ (q − 1)/2 start a new block M λ foreach τ ∈ W λ from (13) foreach σ ∈ W λ from (13) compute p τ ,σ from (20) , one must set all variables z(ω) with ω ∈ Ω 3 an orbit corresponding to a code of minimum Lee (respectively, Lee ∞ ) distance < d to zero. If rows and columns in matrix blocks M λ consist only of zeros after the replacement, it is useful to remove these rows and columns.…”

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confidence: 99%

Semidefinite programming bounds for Lee codes

Polak

2019

Discrete Mathematics

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For q, n, d ∈ N, let A L q (n, d) denote the maximum cardinality of a code C ⊆ Z n q with minimum Lee distance at least d, where Z q denotes the cyclic group of order q. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on A L q (n, d). The technique also yields an upper bound on the independent set number of the n-th strong product power of the circular graph C d,q , which number is related to the Shannon capacity of C d,q . Here C d,q is the graph with vertex set Z q , in which two vertices are adjacent if and only if their distance (mod q) is strictly less than d. The new bound does not seem to improve significantly over the bound obtained from Lovász theta-function, except for very small n.

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“…For the Hamming metric, extensive tables of bounds on the size of codes for both binary and non-binary codes have been constructed, and such tables can be found, for example, in [17]. For the Lee metric, some values have been computed for small alphabets in [18]. In [19], a recursive formula for computing the Lee-numbers needed in the computation of the linear programming bound was introduced and more extensive tables with presently known best upper bounds can be found in [20].…”

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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New upper bounds on Lee codes

Cited by 10 publications

References 13 publications

Generalized upper bounds on the minimum distance of PSK block codes

Generalized upper bounds on the minimum distance of PSK block codes

Semidefinite programming bounds for Lee codes

On the linear programming bound for linear Lee codes

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