Abstract

Polar codes, introduced by Arikan, achieve the capacity of arbitrary binary-input discrete memoryless channel W under successive cancellation decoding. Any such channel having capacity I(W) and for any coding scheme allowing transmission at rate R, scaling exponent is a parameter which characterizes how fast gap to capacity decreases as a function of code length N for a fixed probability of error. The relation between them is given by N⩾α/(I(W)−R)μ. Scaling exponent for kernels of small size up to L=8 have been exhaustively found. In this paper, we consider product kernels TL obtained by taking Kronecker product of component kernels. We derive the properties of polarizing product kernels relating to number of product kernels, self duality and partial distances in terms of the respective properties of the smaller component kernels. Subsequently, polarization behavior of component kernel Tl is used to calculate scaling exponent of TL=T2⊗Tl. Using this method, we show that μ(T2⊗T5)=3.942. Further, we employ a heuristic approach to construct good kernel of L=14 from kernel having size l=8 having best μ and find μ(T2⊗T7)=3.485.

Abstract

Maximally recoverable codes are a class of codes which recover from all potentially recoverable erasure patterns given the locality constraints of the code. In earlier works, these codes have been studied in the context of codes with locality. The notion of locality has been extended to hierarchical locality, which allows for locality to gradually increase in levels with the increase in the number of erasures. We consider the locality constraints imposed by codes with two-level hierarchical locality and define maximally recoverable codes with data-local and local hierarchical locality. We derive certain properties related to their punctured codes and minimum distance. We give a procedure to construct hierarchical data-local MRCs from hierarchical local MRCs. We provide a construction of hierarchical local MRCs for all parameters. For the case of one global parity, we provide a different construction of hierarchical local MRC over a lower field size

Abstract

Cooperative regenerating codes are regenerating codes designed to tradeoff storage for repair bandwidth in case of multiple node failures. Minimum storage cooperative regenerating (MSCR) codes are a class of cooperative regenerating codes which achieve the minimum storage point of the tradeoff. Recently, these codes have been constructed for all possible parameters (n, k, d, h), where h erasures are repaired by contacting any d surviving nodes. However, these constructions have very large sub-packetization. ε-MSR codes are a class of codes introduced to tradeoff subpacketization level for a slight increase in the repair bandwidth for the case of single node failures. We introduce the framework of ε-MSCR codes which allow for a similar tradeoff for the case of multiple node failures. We present a construction of ε-MSCR codes, which can recover from two node failures, by concatenating a class of MSCR codes and scalar linear codes. We give a repair procedure to repair the ε-MSCR codes in the event of two node failures and calculate the repair bandwidth for the same. We characterize the increase in repair bandwidth incurred by the method in comparison with the optimal repair bandwidth given by the cut-set bound. Finally, we show the subpacketization level of ε-MSCR codes scales logarithmically in the number of nodes

Abstract

Autonomous explorative robots frequently encounter scenarios where multiple future trajectories can be pursued. Often these are cases with multiple paths around an obstacle or trajectory options towards various frontiers. Humans in such situations can inherently perceive and reason about the surrounding environment to identify several possibilities of either manoeuvring around the obstacles or moving towards various frontiers. In this work, we propose a 2 stage Convolutional Neural Network architecture which mimics such an ability to map the perceived surroundings to multiple trajectories that a robot can choose to traverse. The first stage is a Trajectory Proposal Network which suggests diverse regions in the environment which can be occupied in the future. The second stage is a Trajectory Sampling network which provides a finegrained trajectory over the regions proposed by Trajectory Proposal Network. We evaluate our framework in diverse and complicated real life settings. For the outdoor case, we use the KITTI dataset and our own outdoor driving dataset. In the indoor setting, we use an autonomous drone to navigate various scenarios and also a ground robot which can explore the environment using the trajectories proposed by our framework. Our experiments suggest that the framework is able to develop a semantic understanding of the obstacles, open regions and identify diverse trajectories that a robot can traverse. Our comparisons portray the performance gain of the proposed architecture over a diverse set of methods against which it is compared.

Abstract

Circular perfect graphs are those undirected graphs such that the circular clique number is equal to the circular chromatic number for each induced subgraph. They form a strict superclass of the perfect graphs, whose index coding broadcast rates are well known. We present the broadcast rate of index coding for side-information graphs whose complements are circular perfect, along with an optimal achievable scheme. We thus enlarge the known classes of graphs for which the broadcast rate is exactly characterized. In an attempt to understand the broadcast rate of a graph given that of its complement, we obtain upper and lower bounds for the product and sum of the vector linear broadcast rates of a graph and its complement. We show that these bounds are satisfied with equality even for some perfect graphs. Curating prior results, we show that there are circular perfect but imperfect graphs which satisfy the lower bound on the product of the broadcast rate of the complementary graphs with equality.

Abstract

Coded caching was introduced as a technique of systematically exploiting locally available storage at the clients to increase the channel throughput via coded transmissions. Most known coded caching schemes in literature enable large gains in terms of the rate, however at the cost of subpacketization that is exponential in ( being the number of clients, some positive integer). Building upon recent prior work for coded caching design via line graphs and finite-field projective geometries, we present a new scheme in this work which achieves a subexponential (in ) subpacketization of and rate , for large , and the cached fraction being upper bounded by a constant (for some prime power and constant ). Apart from this asymptotic improvement, we show that through some numerical comparisons that our present scheme has much lower subpacketization than previous comparable …

Abstract

In this paper, we present IVO: Inverse Velocity Obstacles an egocentric framework that improves the real time implementation. The proposed method stems from the concept of velocity obstacle and can be applied for both single agent and multi-agent system. It focuses on computing collision free maneuvers without any knowledge or assumption on the pose and the velocity of the robot. This is primarily achieved by reformulating the velocity obstacle to adapt to an ego-centric framework. This is a significant step towards improving real time implementations of collision avoidance in dynamic environments as there is no dependency on state estimation techniques to infer the robot pose and velocity. We evaluate IVO for both single agent and multi-agent in different scenarios and show it’s efficacy over the existing formulations. We also show the real time scalability of the proposed methodology.

Abstract

Consider an interference channel consisting of transmitters and receivers with AWGN noise and complex channel gains, and with files in the system. The one-shot for this channel is the maximum number of receivers which can be served simultaneously with vanishing probability of error as the grows large, under a class of schemes known as\textit {one-shot} schemes. Consider that there exists transmitter and receiver side caches which can store fractions and of the library respectively. Recent work for this cache-aided interference channel setup shows that, using a carefully designed prefetching (caching) phase, and a one-shot coded delivery scheme combined with a proper choice of beamforming coefficients at the transmitters, we can achieve a of , where and which was shown to be almost optimal. The existing scheme involves splitting the file into subfiles (the parameter is called the\textit {subpacketization}), where can be extremely large (in fact, with constant cache fractions, it becomes exponential in , for large ). In this work, our first contribution is a scheme which achieves the same of with a smaller subpacketization than prior schemes. Our second contribution is a new coded caching scheme for the interference channel based on projective geometries over finite fields which achieves a one-shot of , with a subpacketization (for some prime power ) that is\textit {subexponential} in , for small constant cache fraction at the receivers. To the best of our knowledge, this is the first coded caching scheme with subpacketization …

Abstract

An index code is said to be locally decodable if each receiver can decode its demand using its side information and by querying only a subset of the transmitted codeword symbols instead of observing the entire codeword. Local decodability can be a beneficial feature in some communication scenarios, such as when the receivers can afford to listen to only a part of the transmissions because of limited availability of power. The locality of an index code is the ratio of the maximum number of codeword symbols queried by a receiver to the message length. In this paper we analyze the optimum locality of linear codes for the family of index coding problems whose min-rank is one less than the number of receivers in the network. We first derive the optimal trade-off between the index coding rate and locality with vector linear coding when the side information graph is a directed cycle. We then provide the optimal trade-off …

Abstract

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called cache) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. In this work, we propose a binary matrix model to construct the coded caching scheme. The ones in such a caching matrix indicate uncached subfiles at the users. Identity submatrices of the caching matrix represent transmissions in the delivery phase. Using this model, we then propose several novel constructions for coded caching based on the various types of combinatorial designs …