Cycles in Oriented 3-graphs

An oriented 3-graph consists of a family of triples (3-sets), each of which is given one of its two possible cyclic orientations. A cycle in an oriented 3-graph is a positive sum of some of the triples that gives weight zero to each 2-set. Our aim in this paper is to consider the following question: how large can the girth of an oriented 3-graph (on $n$ vertices) be? We show that there exist oriented 3-graphs whose shortest cycle has length $\frac{n^2}{2}(1+o(1))$: this is asymptotically best possible. We also show that there exist 3-tournaments whose shortest cycle has length $\frac{n^2}{3}(1+o(1))$, in complete contrast to the case of 2-tournaments.Comment: 12 page

Improved Bounds for the Graham-Pollak Problem for Hypergraphs

For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-graphs needed to partition the complete $r$-graph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. An easy construction shows that $f_r(n) \leq (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$, and we write $c_r$ for the least number such that $f_r(n) \leq c_r (1+o(1))\binom{n}{\lfloor r/2 \rfloor}$. It was known that $c_r < 1$ for each even $r \geq 4$, but this was not known for any odd value of $r$. In this short note, we prove that $c_{295}<1$. Our method also shows that $c_r \rightarrow 0$, answering another open problem

Iterative Multiuser Minimum Symbol Error Rate Beamforming Aided QAM Receiver

A novel iterative soft interference cancellation (SIC) aided beamforming receiver is developed for high-throughput quadrature amplitude modulation systems. The proposed SIC based minimum symbol error rate (MSER) multiuser detection scheme guarantees the direct and explicit minimization of the symbol error rate at the output of the detector. Adopting the extrinsic information transfer (EXIT) chart technique, we compare the EXIT characteristics of an iterative MSER multiuser detector (MUD) with those of the conventional minimum mean-squared error (MMSE) detector. As expected, the proposed SIC-MSER MUD outperforms the SIC-MMSE MUD. Index Terms—Beamforming, iterative multiuser detection, minimum symbol error rate, quadrature amplitude modulation

Symmetric complex-valued RBF receiver for multiple-antenna aided wireless systems

A nonlinear beamforming assisted detector is proposed for multiple-antenna-aided wireless systems employing complex-valued quadrature phase shift-keying modulation. By exploiting the inherent symmetry of the optimal Bayesian detection solution, a novel complex-valued symmetric radial basis function (SRBF)-network-based detector is developed, which is capable of approaching the optimal Bayesian performance using channel-impaired training data. In the uplink case, adaptive nonlinear beamforming can be efficiently implemented by estimating the system’s channel matrix based on the least squares channel estimate. Adaptive implementation of nonlinear beamforming in the downlink case by contrast is much more challenging, and we adopt a cluster-variationenhanced clustering algorithm to directly identify the SRBF center vectors required for realizing the optimal Bayesian detector. A simulation example is included to demonstrate the achievable performance improvement by the proposed adaptive nonlinear beamforming solution over the theoretical linear minimum bit error rate beamforming benchmark