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Register a new userAn Explicit Rate-Optimal Streaming Code for Channels with Burst and Arbitrary Erasures
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Added by
Elad Domanovitz
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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This paper considers the transmission of an infinite sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. While the capacity of a channel that introduces only bursts of erasures is well known, only recently, the capacity of a channel with either one burst of erasures or multiple arbitrary erasures in any fixed-sized sliding window has been established. However, the codes shown to achieve this capacity are either non-explicit constructions (proven to exist) or explicit constructions that require large field size that scales exponentially with the delay. This work describes an explicit rate-optimal construction for admissible channel and delay parameters over a field size that scales only quadratically with the delay.
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Streaming codes are a class of packet-level erasure codes that ensure packet recovery over a sliding window channel which allows either a burst erasure of size $b$ or $a$ random erasures within any window of size $(tau+1)$ time units, under a strict decoding-delay constraint $tau$. The field size over which streaming codes are constructed is an important factor determining the complexity of implementation. The best known explicit rate-optimal streaming code requires a field size of $q^2$ where $q ge tau+b-a$ is a prime power. In this work, we present an explicit rate-optimal streaming code, for all possible ${a,b,tau}$ parameters, over a field of size $q^2$ for prime power $q ge tau$. This is the smallest-known field size of a general explicit rate-optimal construction that covers all ${a,b,tau}$ parameter sets. We achieve this by modifying the non-explicit code construction due to Krishnan et al. to make it explicit, without change in field size.
Achievable rate regions for cooperative relay broadcast channels with rate-limited feedback are proposed. Specifically, we consider two-receiver memoryless broadcast channels where each receiver sends feedback signals to the transmitter through a noiseless and rate-limited feedback link, and meanwhile, acts as relay to transmit cooperative information to the other receiver. Its shown that the proposed rate regions improve on the known regions that consider either relaying cooperation or feedback communication, but not both.
The Welch Bound is a lower bound on the root mean square cross correlation between $n$ unit-norm vectors $f_1,...,f_n$ in the $m$ dimensional space ($mathbb{R} ^m$ or $mathbb{C} ^m$), for $ngeq m$. Letting $F = [f_1|...|f_n]$ denote the $m$-by-$n$ frame matrix, the Welch bound can be viewed as a lower bound on the second moment of $F$, namely on the trace of the squared Gram matrix $(FF)^2$. We consider an erasure setting, in which a reduced frame, composed of a random subset of Bernoulli selected vectors, is of interest. We extend the Welch bound to this setting and present the {em erasure Welch bound} on the expected value of the Gram matrix of the reduced frame. Interestingly, this bound generalizes to the $d$-th order moment of $F$. We provide simple, explicit formulae for the generalized bound for $d=2,3,4$, which is the sum of the $d$-th moment of Wachters classical MANOVA distribution and a vanishing term (as $n$ goes to infinity with $frac{m}{n}$ held constant). The bound holds with equality if (and for $d = 4$ only if) $F$ is an Equiangular Tight Frame (ETF). Our results offer a novel perspective on the superiority of ETFs over other frames in a variety of applications, including spread spectrum communications, compressed sensing and analog coding.
Non-malleable codes protect against an adversary who can tamper with the coded message by using a tampering function in a specified function family, guaranteeing that the tampering result will only depend on the chosen function and not the coded message. The codes have been motivated for providing protection against tampering with hardware that stores the secret cryptographic keys, and have found significant attention in cryptography. Traditional Shannon model of communication systems assumes the communication channel is perfectly known to the transmitter and the receiver. Arbitrary Varying Channels (AVCs) remove this assumption and have been used to model adversarially controlled channels. Transmission over these channels has been originally studied with the goal of recovering the sent message, and more recently with the goal of detecting tampering with the sent messages. In this paper we introduce non-malleability as the protection goal of message transmission over these channels, and study binary (discrete memoryless) AVCs where possible tampering is modelled by the set of channel states. Our main result is that non-malleability for these channels is achievable at a rate asymptotically approaching 1. We also consider the setting of an AVC with a special state s*, and the additional requirement that the message must be recoverable if s* is applied to all the transmitted bits. We give the outline of a message encoding scheme that in addition to non-malleability, can provide recovery for all s* channel.
A partially cooperative relay broadcast channel (RBC) is a three-node network with one source node and two destination nodes (destinations 1 and 2) where destination 1 can act as a relay to assist destination 2. Inner and outer bounds on the capacity region of the discrete memoryless partially cooperative RBC are obtained. When the relay function is disabled, the inner and outer bounds reduce to new bounds on the capacity region of broadcast channels. Four classes of RBCs are studied in detail. For the partially cooperative RBC with degraded message sets, inner and outer bounds are obtained. For the semideterministic partially cooperative RBC and the orthogonal partially cooperative RBC, the capacity regions are established. For the parallel partially cooperative RBC with unmatched degraded subchannels, the capacity region is established for the case of degraded message sets. The capacity is also established when the source node has only a private message for destination 2, i.e., the channel reduces to a parallel relay channel with unmatched degraded subchannels.
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