Optimal conflict-avoiding codes of odd length and weight three

Fu, Hung‐Lin; Lo, Yuan-Hsun; Shum, Kenneth W.

doi:10.1007/s10623-012-9764-5

Cited by 22 publications

(31 citation statements)

References 15 publications

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“…It is a difficult problem to determine the exact value of

{normalΨ}^{e} false(m, 3, 2, 1 false)

for all m . Actually when

m ≢ 0 (mod 3)

{normalΨ}^{e} false(m, 3, 2, 1 false) = {normalΨ}^{e} false(m, 3, 3, 1 false)

, and there have been many papers concerning the estimation of

{normalΨ}^{e} false(m, 3, 3, 1 false)

under the background of conflict avoiding codes.…”

Section: Resultsmentioning

confidence: 99%

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Feng

Wang

et al. 2016

J of Combinatorial Designs

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The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation λa and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional (n×m,k,λa,1)‐optical orthogonal codes. When k=3 and λa=2, the exact number of codewords of an optimal two‐dimensional (n×m,3,2,1)‐optical orthogonal code is determined for any positive integers n and m≡2(mod4).

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“…It is a difficult problem to determine the exact value of

{normalΨ}^{e} false(m, 3, 2, 1 false)

for all m . Actually when

m ≢ 0 (mod 3)

{normalΨ}^{e} false(m, 3, 2, 1 false) = {normalΨ}^{e} false(m, 3, 3, 1 false)

, and there have been many papers concerning the estimation of

{normalΨ}^{e} false(m, 3, 3, 1 false)

under the background of conflict avoiding codes.…”

Section: Resultsmentioning

confidence: 99%

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Feng

Wang

et al. 2016

J of Combinatorial Designs

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“…It is worth mentioning that a few known results used this idea to obtain optimal (equidifference) conflict-avoiding codes, especially when k = 3 (see [1,2,4]). …”

Section: Proposition 1 the Size Of A Maximum Weighted Matching In G(nmentioning

confidence: 99%

“…In the case of weight k = 3, the characterization of even length was completely settled by [1,4,6,9], and some optimal (equi-difference) CACs of odd length were studied in [2,7,8,10,15]. Several optimal constructions for weight k = 4, 5 can be found in [11].…”

mentioning

confidence: 99%

Weighted maximum matchings and optimal equi-difference conflict-avoiding codes

Lin

2014

Des. Codes Cryptogr.

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A conflict-avoiding code (CAC) C of length n and weight k is a collection of k-subsets of Z n such that Δ(x) ∩ Δ(y) = ∅ for any x, y ∈ C and x = y, where Δ(x) = {a − b : a, b ∈ x, a = b}. Let CAC(n, k) denote the class of all CACs of length n and weight k. A CAC C ∈ CAC(n, k) is said to be equi-difference if any codeword x ∈ C has the form {0, i, 2i, . . . , (k − 1)i}. A CAC with maximum size is called optimal. In this paper we propose a graphical characterization of an equi-difference CAC, and then provide an infinite number of optimal equi-difference CACs for weight four.

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“…

(x, \frac{n - x}{2}))

are contained in

G (Ω_{n})

, this implies that the degree of each vertex in

G (Ω_{n})

is two, and we have the following consequence. It should be mentioned that the graph

G (Ω_{n})

has been used to find the set of equi‐difference codewords in . Lemma For any odd integer n , the graph

G (Ω_{n})

is a union of vertex‐disjoint

m_{i}

‐cycles

(1 \leq i \leq k)

satisfying that

\sum_{i = 1}^{k} m_{i} = \frac{n - 1}{2} .

…”

Section: Constructing Optimal Tight Equi‐difference Codesmentioning

confidence: 99%

Optimal Tight Equi‐Difference Conflict‐Avoiding Codes of Length n = 2^k ± 1 and Weight 3

2012

J of Combinatorial Designs

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For a k-subset X of Z n , the set of differences on X is the set X = {i − j (mod n): i, j ∈ X, i = j } . A conflict-avoiding code CAC of length n and weight k is a collection C of k-subsets of Z n such that X Y = ∅ for any distinct X, Y ∈ C. Let CAC(n, k) be the class of all the CACs of length n and weight k. The maximum size of codes in CAC(n, k) is denoted by M(n, k). A code C ∈ CAC(n, k) is said to be optimal if |C| = M(n, k). An optimal code C is tight equi-difference if X∈C X = Z n \ {0} and each codeword in C is of the form {0, i, 2i, . . . , (k − 1)i}. In this paper, the necessary and sufficient conditions for the existence problem of optimal tight equi-difference conflict-avoiding codes of length n = 2 k ± 1 and weight 3 are given. C 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 223-231, 2013

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Optimal conflict-avoiding codes of odd length and weight three

Cited by 22 publications

References 15 publications

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Weighted maximum matchings and optimal equi-difference conflict-avoiding codes

Optimal Tight Equi‐Difference Conflict‐Avoiding Codes of Length n = 2^k ± 1 and Weight 3

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Optimal conflict-avoiding codes of odd length and weight three

Cited by 22 publications

References 15 publications

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

Weighted maximum matchings and optimal equi-difference conflict-avoiding codes

Optimal Tight Equi‐Difference Conflict‐Avoiding Codes of Length n = 2k ± 1 and Weight 3

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Optimal Tight Equi‐Difference Conflict‐Avoiding Codes of Length n = 2^k ± 1 and Weight 3