On the Number of Latin Rectangles

Stones, Douglas S.

doi:10.1017/s0004972710000328

Cited by 23 publications

(44 citation statements)

References 15 publications

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“…The following theorem also incidentally gives a divisor for R n , which sometimes improves on Theorem 3.3 (see [21] for more about R n ). …”

Section: Theorem 32 Identifies That R Ementioning

confidence: 72%

Hownotto prove the Alon-Tarsi conjecture

Stones¹,

Wanless²

2012

Nagoya Mathematical Journal

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Abstract. The sign of a Latin square is −1 if it has an odd number of rows and columns that are odd permutations; otherwise, it is +1. Let L e n and L o n be, respectively, the number of Latin squares of order n with sign +1 and −1.3 ) for prime p ≥ 3 and asked if similar congruences hold for orders of the form p k + 1, p + 3, or pq + 1. In this article we show) only if t = n and n is an odd prime, thereby showing that Drisko's method cannot be extended to encompass any of the three suggested cases. We also extend exact computation to n ≤ 9, discuss asymptotics for L o /L e , and propose a generalization of the Alon-Tarsi conjecture.

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“…The following theorem also incidentally gives a divisor for R n , which sometimes improves on Theorem 3.3 (see [21] for more about R n ). …”

Section: Theorem 32 Identifies That R Ementioning

confidence: 72%

Hownotto prove the Alon-Tarsi conjecture

Stones¹,

Wanless²

2012

Nagoya Mathematical Journal

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“…Definition 4: [13] Two Latin Squares L and L (using the same symbol set) are isotopic if there is a triple (f,g,h), where f is a row permutation, g is a column permutation and h is a symbol permutation, such that applying these permutations on L gives L .…”

Section: A Removing Singular Fade States Singularity-removal Constrmentioning

confidence: 99%

Wireless Network-Coded Bidirectional Relaying Using Latin Squares for <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-PSK Modulation

Muralidharan

Namboodiri

Rajan

2013

IEEE Trans. Inform. Theory

106

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Abstract-The design of modulation schemes for the physical layer network-coded two way relaying scenario is considered with a protocol which employs two phases: Multiple access (MA) phase and Broadcast (BC) phase. It was observed by KoikeAkino et. al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, that this relationship can be used to get the network coding maps satisfying the exclusive law. The channel fade states for which the minimum distance of the effective constellation at the relay become zero are referred to as the singular fade states. For M −PSK modulation (M any power of 2), it is shown that there are V. Namboodiri is currently with Qualcomm India Private Limited, Hyderabad-500081, India (e-mail: namboodiri.vishnu@gmail.com). The work was carried out when he was with the Dept. of Electrical Communication Engineering, Indian Institute of Science, Bangalore-560012, India.Communicated by J.-C. Belfiore, Associate Editor for Coding Theory. I. BACKGROUND AND PRELIMINARIESWe consider the two-way wireless relaying scenario shown in Fig. 1, where bidirectional data transfer takes place between the nodes A and B with the help of the relay R. It is assumed that all three nodes operate in half-duplex mode, i.e., they cannot transmit and receive simultaneously in the same frequency band. The relaying protocol consists of the following two phases: the multiple access (MA) phase, during which A and B simultaneously transmit to R and the broadcast (BC) phase during which R transmits to A and B. Network coding is employed at R in such a way that A (B) can decode the message of B (A), given that A (B) knows its own message. The concept of physical layer network coding has attracted a lot of attention in recent times. The idea of physical layer network coding for the two way relay channel was first introduced in [3], where the multiple access interference occurring at the relay was exploited so that the communication between the end nodes can be done using a two phase protocol. Information theoretic studies for the physical layer network coding scenario were reported in [4], [5]. A differential modulation scheme with analog network coding for bidirectional relaying was proposed in [6]. The design principles governing the choice of modulation schemes to be used at the nodes for uncoded transmission were studied in [7]. An extension for the case when the nodes use convolutional codes was done in [8]. A multi-level coding scheme for the two-way relaying scenario was proposed in [9]. Power allocation strategies and lattice based coding schemes for bi-directional relaying were proposed in [10].It was observed in [7] that for uncoded transmission, the network c...

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“…A table of values is below, calculated from the exact quantities and rounded to a few significant digits; the values for |LS n | can be found in the recent survey [28]. In addition, we calculate the asymptotic rate of increase of E 1/n 2 n in Proposition 5.5.…”

Section: Theorem 54 Algorithm 3 Samples Uniformly Over the Set Of Amentioning

confidence: 99%

Exact Sampling Algorithms for Latin Squares and Sudoku Matrices via Probabilistic Divide-and-Conquer

DeSalvo

2016

Algorithmica

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Abstract. We provide several algorithms for the exact, uniform random sampling of Latin squares and Sudoku matrices via probabilistic divide-and-conquer (PDC). Our approach divides the sample space into smaller pieces, samples each separately, and combines them in a manner which yields an exact sample from the target distribution. We demonstrate, in particular, a version of PDC in which one of the pieces is sampled using a brute force approach, which we dub almost deterministic second half, as it is a generalization to a previous application of PDC for which one of the pieces is uniquely determined given the others.

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On the Number of Latin Rectangles

Cited by 23 publications

References 15 publications

Hownotto prove the Alon-Tarsi conjecture

Hownotto prove the Alon-Tarsi conjecture

Wireless Network-Coded Bidirectional Relaying Using Latin Squares for <formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula>-PSK Modulation

Exact Sampling Algorithms for Latin Squares and Sudoku Matrices via Probabilistic Divide-and-Conquer

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