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Register a new userStreaming Erasure Codes over the Multiple Access Relay Channel
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Added by
Gustavo Kasper Facenda
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Applications where multiple users communicate with a common server and desire low latency are common and increasing. This paper studies a network with two source nodes, one relay node and a destination node, where each source nodes wishes to transmit a sequence of messages, through the relay, to the destination, who is required to decode the messages with a strict delay constraint $T$. The network with a single source node has been studied in cite{Silas2019}. We start by introducing two important tools: the delay spectrum, which generalizes delay-constrained point-to-point transmission, and concatenation, which, similar to time sharing, allows combinations of different codes in order to achieve a desired regime of operation. Using these tools, we are able to generalize the two schemes previously presented in cite{Silas2019}, and propose a novel scheme which allows us to achieve optimal rates under a set of well-defined conditions. Such novel scheme is further optimized in order to improve the achievable rates in the scenarios where the conditions for optimality are not met.
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This paper studies low-latency streaming codes for the multi-hop network. The source is transmitting a sequence of messages (streaming messages) to a destination through a chain of relays where each hop is subject to packet erasures. Every source message has to be recovered perfectly at the destination within a delay constraint of $T$ time slots. In any sliding window of $T+1$ time slots, we assume no more than $N_j$ erasures introduced by the $j$th hop channel. The capacity in case of a single relay (a three-node network) was derived by Fong [1], et al. While the converse derived for the three-node case can be extended to any number of nodes using a similar technique (analyzing the case where erasures on other links are consecutive), we demonstrate next that the achievable scheme, which suggested a clever symbol-wise decode and forward strategy, can not be straightforwardly extended without a loss in performance. The coding scheme for the three-node network, which was shown to achieve the upper bound, was ``state-independent (i.e., it does not depend on specific erasure pattern). While this is a very desirable property, in this paper, we suggest a ``state-dependent (i.e., a scheme which depends on specific erasure pattern) and show that it achieves the upper bound up to the size of an additional header. Since, as we show, the size of the header does not depend on the field size, the gap between the achievable rate and the upper bound decreases as the field size increases.
This paper studies the problem of secure communication over a K-transmitter multiple access channel in the presence of an external eavesdropper, subject to a joint secrecy constraint (i.e., information leakage rate from the collection of K messages to an eavesdropper is made vanishing). As a result, we establish the joint secrecy achievable rate region. To this end, our results build upon two techniques in addition to the standard information-theoretic methods. The first is a generalization of Chia-El Gamals lemma on entropy bound for a set of codewords given partial information. The second is to utilize a compact representation of a list of sets that, together with properties of mutual information, leads to an efficient Fourier-Motzkin elimination. These two approaches could also be of independent interests in other contexts.
Streaming codes are a class of packet-level erasure codes that are designed with the goal of ensuring recovery in low-latency fashion, of erased packets over a communication network. It is well-known in the streaming code literature, that diagonally embedding codewords of a $[tau+1,tau+1-a]$ Maximum Distance Separable (MDS) code within the packet stream, leads to rate-optimal streaming codes capable of recovering from $a$ arbitrary packet erasures, under a strict decoding delay constraint $tau$. Thus MDS codes are geared towards the efficient handling of the worst-case scenario corresponding to the occurrence of $a$ erasures. In the present paper, we have an increased focus on the efficient handling of the most-frequent erasure patterns. We study streaming codes which in addition to recovering from $a>1$ arbitrary packet erasures under a decoding delay $tau$, have the ability to handle the more common occurrence of a single-packet erasure, while incurring smaller delay $r<tau$. We term these codes as $(a,tau,r)$ locally recoverable streaming codes (LRSCs), since our single-erasure recovery requirement is similar to the requirement of locality in a coded distributed storage system. We characterize the maximum possible rate of an LRSC by presenting rate-optimal constructions for all possible parameters ${a,tau,r}$. Although the rate-optimal LRSC construction provided in this paper requires large field size, the construction is explicit. It is also shown that our $(a,tau=a(r+1)-1,r)$ LRSC construction provides the additional guarantee of recovery from the erasure of $h, 1 leq h leq a$, packets, with delay $h(r+1)-1$. The construction thus offers graceful degradation in decoding delay with increasing number of erasures.
The Gilbert-Elliot (GE) channel is a commonly-accepted model for packet erasures in networks. Streaming codes are a class of packet-level erasure codes designed to provide reliable communication over the GE channel. The design of a streaming code may be viewed as a two-step process. In the first, a more tractable, delay-constrained sliding window (DCSW) channel model is considered as a proxy to the GE channel. The streaming code is then designed to reliably recover from all erasures introduced by the DCSW channel model. Simulation is typically used to evaluate the performance of the streaming code over the original GE channel, as analytic performance evaluation is challenging. In the present paper, we take an important first step towards analytical performance evaluation. Recognizing that most, efficient constructions of a streaming code are based on the diagonal embedding or horizontal embedding of scalar block codes within a packet stream, this paper provides upper and lower bounds on the block-erasure probability of the underlying scalar block code when operated over the GE channel.
A lower bound on the maximum likelihood (ML) decoding error exponent of linear block code ensembles, on the erasure channel, is developed. The lower bound turns to be positive, over an ensemble specific interval of erasure probabilities, when the ensemble weight spectral shape function tends to a negative value as the fractional codeword weight tends to zero. For these ensembles we can therefore lower bound the block-wise ML decoding threshold. Two examples are presented, namely, linear random parity-check codes and fixed-rate Raptor codes with linear random precoders. While for the former a full analytical solution is possible, for the latter we can lower bound the ML decoding threshold on the erasure channel by simply solving a 2 x 2 system of nonlinear equations.
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