50 Years of the Golomb--Welch Conjecture

Since 1968, when the Golomb--Welch conjecture was raised, it has become the main motive power behind the progress in the area of the perfect Lee codes. Although there is a vast literature on the topic and it is widely believed to be true, this conjecture is far from being solved. In this paper, we provide a survey of papers on the Golomb--Welch conjecture. Further, new results on Golomb--Welch conjecture dealing with perfect Lee codes of large radii are presented. Algebraic ways of tackling the conjecture in the future are discussed as well. Finally, a brief survey of research inspired by the conjecture is given.Comment: 28 pages, 2 figure

Relativistic models of magnetars: Nonperturbative analytical approach

In the present paper we focus on building simple nonperturbative analytical relativistic models of magnetars. With this purpose in mind we first develop a method for generating exact interior solutions to the static and axisymmetric Einstein-Maxwell-hydrodynamic equations with anisotropic perfect fluid and with pure poloidal magnetic field. Then using an explicit exact solution we present a simple magnetar model and calculate some physically interesting quantities as the surface elipticity and the total energy of the magnetized star.Comment: 10 pages, LaTe

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Decentralized Estimation over Orthogonal Multiple-access Fading Channels in Wireless Sensor Networks - Optimal and Suboptimal Estimators

Optimal and suboptimal decentralized estimators in wireless sensor networks (WSNs) over orthogonal multiple-access fading channels are studied in this paper. Considering multiple-bit quantization before digital transmission, we develop maximum likelihood estimators (MLEs) with both known and unknown channel state information (CSI). When training symbols are available, we derive a MLE that is a special case of the MLE with unknown CSI. It implicitly uses the training symbols to estimate the channel coefficients and exploits the estimated CSI in an optimal way. To reduce the computational complexity, we propose suboptimal estimators. These estimators exploit both signal and data level redundant information to improve the estimation performance. The proposed MLEs reduce to traditional fusion based or diversity based estimators when communications or observations are perfect. By introducing a general message function, the proposed estimators can be applied when various analog or digital transmission schemes are used. The simulations show that the estimators using digital communications with multiple-bit quantization outperform the estimator using analog-and-forwarding transmission in fading channels. When considering the total bandwidth and energy constraints, the MLE using multiple-bit quantization is superior to that using binary quantization at medium and high observation signal-to-noise ratio levels

Two-tier channel estimation aided near-capacity MIMO transceivers relying on norm-based joint transmit and receive antenna selection

We propose a norm-based joint transmit and receive antenna selection (NBJTRAS) aided near-capacity multiple-input multiple-output (MIMO) system relying on the assistance of a novel two-tier channel estimation scheme. Specifically, a rough estimate of the full MIMO channel is first generated using a low-complexity, low-training-overhead minimum mean square error based channel estimator, which relies on reusing a modest number of radio frequency (RF) chains. NBJTRAS is then carried out based on this initial full MIMO channel estimate. The NBJTRAS aided MIMO system is capable of significantly outperforming conventional MIMO systems equipped with the same modest number of RF chains, while dispensing with the idealised simplifying assumption of having perfectly known channel state information (CSI). Moreover, the initial subset channel estimate associated with the selected subset MIMO channel matrix is then used for activating a powerful semi-blind joint channel estimation and turbo detector-decoder, in which the channel estimate is refined by a novel block-of-bits selection based soft-decision aided channel estimator (BBSB-SDACE) embedded in the iterative detection and decoding process. The joint channel estimation and turbo detection-decoding scheme operating with the aid of the proposed BBSB-SDACE channel estimator is capable of approaching the performance of the near-capacity maximumlikelihood (ML) turbo transceiver associated with perfect CSI. This is achieved without increasing the complexity of the ML turbo detection and decoding process

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure