Multicast Network Coding and Field Sizes

In an acyclic multicast network, it is well known that a linear network coding solution over GF($q$) exists when $q$ is sufficiently large. In particular, for each prime power $q$ no smaller than the number of receivers, a linear solution over GF($q$) can be efficiently constructed. In this work, we reveal that a linear solution over a given finite field does \emph{not} necessarily imply the existence of a linear solution over all larger finite fields. Specifically, we prove by construction that: (i) For every source dimension no smaller than 3, there is a multicast network linearly solvable over GF(7) but not over GF(8), and another multicast network linearly solvable over GF(16) but not over GF(17); (ii) There is a multicast network linearly solvable over GF(5) but not over such GF($q$) that $q > 5$ is a Mersenne prime plus 1, which can be extremely large; (iii) A multicast network linearly solvable over GF($q^{m_1}$) and over GF($q^{m_2}$) is \emph{not} necessarily linearly solvable over GF($q^{m_1+m_2}$); (iv) There exists a class of multicast networks with a set $T$ of receivers such that the minimum field size $q_{min}$ for a linear solution over GF($q_{min}$) is lower bounded by $\Theta(\sqrt{|T|})$, but not every larger field than GF($q_{min}$) suffices to yield a linear solution. The insight brought from this work is that not only the field size, but also the order of subgroups in the multiplicative group of a finite field affects the linear solvability of a multicast network

Full-duplex wireless communications: challenges, solutions and future research directions

The family of conventional half-duplex (HD) wireless systems relied on transmitting and receiving in different time-slots or frequency sub-bands. Hence the wireless research community aspires to conceive full-duplex (FD) operation for supporting concurrent transmission and reception in a single time/frequency channel, which would improve the attainable spectral efficiency by a factor of two. The main challenge encountered in implementing an FD wireless device is the large power difference between the self-interference (SI) imposed by the device’s own transmissions and the signal of interest received from a remote source. In this survey, we present a comprehensive list of the potential FD techniques and highlight their pros and cons. We classify the SI cancellation techniques into three categories, namely passive suppression, analog cancellation and digital cancellation, with the advantages and disadvantages of each technique compared. Specifically, we analyse the main impairments (e.g. phase noise, power amplifier nonlinearity as well as in-phase and quadrature-phase (I/Q) imbalance, etc.) that degrading the SI cancellation. We then discuss the FD based Media Access Control (MAC)-layer protocol design for the sake of addressing some of the critical issues, such as the problem of hidden terminals, the resultant end-to-end delay and the high packet loss ratio (PLR) due to network congestion. After elaborating on a variety of physical/MAC-layer techniques, we discuss potential solutions conceived for meeting the challenges imposed by the aforementioned techniques. Furthermore, we also discuss a range of critical issues related to the implementation, performance enhancement and optimization of FD systems, including important topics such as hybrid FD/HD scheme, optimal relay selection and optimal power allocation, etc. Finally, a variety of new directions and open problems associated with FD technology are pointed out. Our hope is that this treatise will stimulate future research efforts in the emerging field of FD communication