The sum degrees of freedom (DoF) of the two-transmitter, two-receiver multiple-input multipleoutput (MIMO) X-Network (2 × 2 MIMO X-Network) with M antennas at each node is known to be 4M 3 . Transmission schemes which couple local channel-state-information-at-the-transmitter (CSIT) based precoding with space-time block coding to achieve the sum-DoF of this network are known specifically for M = 2, 4. These schemes have been proven to guarantee a diversity gain of M when a finite-sized input constellation is employed. In this paper, an explicit transmission scheme that achieves the 4M 3 sum-DoF of the 2 × 2 X-Network for arbitrary M is presented. The proposed scheme needs only local CSIT unlike the Jafar-Shamai scheme which requires the availability of global CSIT in order to achieve the 4M 3 sum-DoF. Further, it is shown analytically that the proposed scheme guarantees a diversity gain of M + 1 when finite-sized input constellations are employed.Freedom.that cell edge users are susceptible to interference from the neighbouring base stations and vice-versa.These issues have instigated research on better transmission techniques in interference networks, with information-theoretic rate tuples often used as the metric for designing better schemes. Since the capacity of interference networks is unknown in general, degrees of freedom (DoF) [1] is the widely targeted metric due to its relative ease of characterization. The sum-DoF of a Gaussian network is said to be d if its sum-capacity (in bits per channel use) can be approximated as C(SNR) = d log 2 SNR + o(log 2 SNR).Availability of channel-state-information at the transmitters (CSIT) is an important assumption in the characterization of the approximate capacity of Gaussian interference networks. Availability of perfect global CSIT 1 often enables one to design precoders that cast interference onto subspaces independent of the desired signal space at the receivers. This technique, termed interference alignment (IA), was first used implicitly in [2], [3], and explicitly appeared in [4], [5] in the context of 2 × 2 multiple-input multiple-output (MIMO) X-Networks. A K × J X-Network is a Gaussian interference network with K transmitters and J receivers and a total of KJ independent messages meant to be sent over the network, one from every transmitter to every receiver. A 2×2 X-Network with M antennas at each node is referred to as the (2 × 2, M ) X-Network. A lower bound on the sum-DoF was shown to be 4M 3 for such a network in [3], and it was proven in [5] that the sum-DoF equals 4M 3 , achieved using an IA scheme. All the aforementioned works assume the availability of perfect global CSIT.The concept of DoF assumes the use of a codebook with unconstrained alphabet size as well as unlimited peak power, but with an average power constraint. The channel is assumed to be static during the transmission of an entire codeword. Further, information-theoretic rate definitions also assume the usage of unlimited coding length. Clearly, all these assumptions are infeasible in p...

Universally decodable matrices (UDMs) can be used for coding purposes when transmitting over slow fading channels. These matrices are parameterized by positive integers L and N and a prime power q. The main result of this paper is that the simple condition L ≤ q + 1 is both necessary and sufficient for (L, N, q)-UDMs to exist. The existence proof is constructive and yields a coding scheme that is equivalent to a class of codes that was proposed by Rosenbloom and Tsfasman. Our work resolves an open problem posed recently in the literature.

The complete transposition graph is defined to be the graph whose vertices are the elements of the symmetric group $S_n$, and two vertices $\alpha$ and $\beta$ are adjacent in this graph iff there is some transposition $(i,j)$ such that $\alpha=(i,j) \beta$. Thus, the complete transposition graph is the Cayley graph $\Cay(S_n,S)$ of the symmetric group generated by the set $S$ of all transpositions. An open problem in the literature is to determine which Cayley graphs are normal. It was shown recently that the Cayley graph generated by 4 cyclically adjacent transpositions is not normal. In the present paper, it is proved that the complete transposition graph is not a normal Cayley graph, for all $n \ge 3$. Furthermore, the automorphism group of the complete transposition graph is shown to equal \[ \Aut(\Cay(S_n,S)) = (R(S_n) \rtimes \Inn(S_n)) \rtimes \mathbb{Z}_2, \] where $R(S_n)$ is the right regular representation of $S_n$, $\Inn(S_n)$ is the group of inner automorphisms of $S_n$, and $\mathbb{Z}_2 = \langle h \rangle$, where $h$ is the map $\alpha \mapsto \alpha^{-1}$

Let S be a set of transpositions that generates the symmetric group S n , where n ≥ 3. The transposition graph T (S) is defined to be the graph with vertex set {1, . . . , n} and with vertices i and j being adjacent in T (S) whenever (i, j) ∈ S. We prove that if the girth of the transposition graph T (S) is at least 5, then the automorphism group of the Cayley graph Cay(S n , S) is the semidirect product R(S n )⋊Aut(S n , S), where Aut(S n , S) is the set of automorphisms of S n that fixes S. This strengthens a result of Feng on transposition graphs that are trees. We also prove that if the transposition graph T (S) is a 4-cycle, then the set of automorphisms of the Cayley graph Cay(S 4 , S) that fixes a vertex and each of its neighbors is isomorphic to the Klein 4-group and hence is nontrivial. We thus identify the existence of 4-cycles in the transposition graph as being an important factor in causing a potentially larger automorphism group of the Cayley graph.

Group testing with inhibitors (GTI) introduced by Farach at al. is studied in this paper. There are three types of items, d defectives, r inhibitors and n − d − r normal items in a population of n items. The presence of any inhibitor in a test can prevent the expression of a defective. For this model, we propose a probabilistic non-adaptive pooling design with a low complexity decoding algorithm. We show that the sample complexity of the number of tests required for guaranteed recovery with vanishing error probability using the proposed algorithm scales as T = O(d log n) and T = O( r 2 d log n) in the regimes r = O(d) and d = o(r) respectively. In the former regime, the number of tests meets the lower bound order while in the latter regime, the number of tests is shown to exceed the lower bound order by a log r d multiplicative factor. The decoding complexity of the proposed decoding algorithm scales as O(nT ).