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 $(τ+1)$ time units, under a strict decoding-delay constraint $τ$. 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 $q \ge τ+b-a$ is a prime power. In this work, we present an explicit rate-optimal streaming code, for all possible $\{a,b,τ\}$ parameters, over a field of size $q^2$ for prime power $q \ge τ$. This is the smallest-known field size of a general explicit rate-optimal construction that covers all $\{a,b,τ\}$ 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.
Real-time streaming communication requires a high quality of service despite contending with packet loss. Streaming codes are a class of codes best suited for this setting. A key challenge for streaming codes is that they operate in an "online" setting in which the amount of data to be transmitted varies over time and is not known in advance. Mitigating the adverse effects of variability requires spreading the data that arrives at a time slot over multiple future packets, and the optimal strategy for spreading depends on the arrival pattern. Algebraic coding techniques alone are therefore insu…
This paper considers transmitting a sequence of messages (a streaming source) over a packet erasure channel. In each time slot, the source constructs a packet based on the current and the previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. Every source message must be recovered perfectly at the destination subject to a fixed decoding delay. We assume that the channel loss model introduces either one burst erasure or multiple arbitrary erasures in any fixed-sized sliding window. Under this channel loss assumption, we fully …
This paper presents a new construction of error correcting codes which achieves optimal recovery of a streaming source over a packet erasure channel. The channel model considered is the sliding window erasure model, with burst and arbitrary losses, introduced by Badr et al. . Recently, two independents works by Fong et al. and Krishnan and Kumar have identified optimal streaming codes within this framework. In this paper, we introduce streaming code when the rate of the code is at least 1/2. Our proposed construction is explicit and systematic, uses off-the-shelf maximum distance separable (MD…
Recovery of data packets from packet erasures in a timely manner is critical for many streaming applications. An early paper by Martinian and Sundberg introduced a framework for streaming codes and designed rate-optimal codes that permit delay-constrained recovery from an erasure burst of length up to $B$. A recent work by Badr et al. extended this result and introduced a sliding-window channel model $\mathcal{C}(N,B,W)$. Under this model, in a sliding-window of width $W$, one of the following erasure patterns are possible (i) a burst of length at most $B$ or (ii) at most $N$ (possibly non-con…
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 explici…
One pervasive challenge in providing a high quality-of-service for live communication is to recover lost packets in real-time. Streaming codes are a class of erasure codes that are designed for such strict, low-latency streaming communication settings. Motivated by applications that transmit messages whose sizes vary over time, such as live video streaming, this paper considers the setting of streaming codes under variable-size messages. In practice, streaming codes operate in an "online" setting where the sizes of the future messages are unknown. "Offline" codes, in contrast, have access to t…
An $(a,b,τ)$ streaming code is a packet-level erasure code that can recover under a strict delay constraint of $τ$ time units, from either a burst of $b$ erasures or else of $a$ random erasures, occurring within a sliding window of time duration $w$. While rate-optimal constructions of such streaming codes are available for all parameters $\{a,b,τ,w\}$ in the literature, they require in most instances, a quadratic, $O(τ^2)$ field size. In this work, we make further progress towards field size reduction and present rate-optimal $O(τ)$ field size streaming codes for two regimes: (i) $gcd(b,τ+1-a…
In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$. One needs to recover any erased packet, where erasures are as per the channel model, with a strict decoding delay deadline of $τ$ time slots. Presently existing rate-optimal constructions in the literature require, in general, a field-size which gro…
The explosive demand of on-line video from smart mobile devices poses unprecedented challenges to delivering high quality of experience (QoE) over wireless networks. Streaming high-definition video with low delay is difficult mainly due to (i) the stochastic nature of wireless channels and (ii) the fluctuating videos bit rate. To address this, we propose a novel delay-aware fountain coding (DAF) technique that integrates channel coding and video coding. In this paper, we reveal that the fluctuation of video bit rate can also be exploited to further improve fountain codes for wireless video str…
This paper investigates low-latency streaming codes for a three-node relay network. The source transmits a sequence of messages (streaming messages) to the destination through the relay between them, where the first-hop channel from the source to the relay and the second-hop channel from the relay to the destination are subject to packet erasures. Every source message must be recovered perfectly at the destination subject to a fixed decoding delay of $T$ time slots. In any sliding window of $T+1$ time slots, we assume no more than $N_1$ and $N_2$ erasures are introduced by the first-hop channe…
In this paper, we investigate wireless streaming of multi-quality tiled 360 virtual reality (VR) videos from a multi-antenna server to multiple single-antenna users in a multi-carrier system. To capture the impact of field-of-view (FoV) prediction, we consider three cases of FoV viewing probability distributions, i.e., perfect, imperfect and unknown FoV viewing probability distributions, and use the average total utility, worst average total utility and worst total utility as the respective performance metrics. We adopt rate splitting with successive decoding for efficient transmission of mult…
Streaming codes represent a packet-level FEC scheme for achieving reliable, low-latency communication. In the literature on streaming codes, the commonly-assumed Gilbert-Elliott channel model, is replaced by a more tractable, delay-constrained, sliding-window (DCSW) channel model that can introduce either random or burst erasures. The known streaming codes that are rate optimal over the DCSW channel model are constructed by diagonally embedding a scalar block code across successive packets. These code constructions have field size that is quadratic in the delay parameter $τ$ and have a somewha…
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 $[τ+1,τ+1-a]$ 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 $τ$. Thus MDS codes are geared towards the efficient handling of the worst-case scenario corresponding to the …
We consider a hybrid delivery scheme for streaming content, combining cache-enabled Orthogonal Multipoint Multicast (OMPMC) and on-demand Single-Point Unicast (SPUC) transmissions for heterogeneous networks. The OMPMC service transmits cached files through the whole network to interested users, and users not being satisfied by this service are assigned to the SPUC service to be individually served. The SPUC fetches the requested files from the core network and unicasts them to UEs using cellular beamforming transmissions. We optimize the delivery scheme to minimize the average resource consump…
In this paper, we would like to investigate optimal wireless streaming of a multi-quality tiled 360 virtual reality (VR) video from a server to multiple users. To this end, we propose to maximally exploit potential multicast opportunities by effectively utilizing characteristics of multi-quality tiled 360 VR videos and computation resources at the users' side. In particular, we consider two requirements for quality variation in one field-of-view (FoV), i.e., the absolute smoothness requirement and the relative smoothness requirement, and two video playback modes, i.e., the direct-playback mode…
This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay $T_\mathrm{v}$ and every source message in the second sequence subject to a shorter decoding delay $T_\mathrm{u}\le T_\mathrm{v}$. We assume that the channel loss mo…
In this paper, we study the three-node Decode-and-Forward (D&F) relay network subject to random and burst packet erasures. The source wishes to transmit an infinite stream of packets to the destination via the relay. The three-node D&F relay network is constrained by a decoding delay of T packets, i.e., the packet transmitted by the source at time i must be decoded by the destination by time i+T. For the individual channels from source to relay and relay to destination, we assume a delay-constrained sliding-window (DCSW) based packet-erasure model that can be viewed as a tractable approximatio…
We consider the sequential transmission of a stream of messages over a block-fading multi-input-multi-output (MIMO) channel. A new message arrives at the beginning of each coherence block, and the decoder is required to output each message sequentially, after a delay of $T$ coherence blocks. In the special case when $T=1$, the setup reduces to the quasi-static fading channel. We establish the optimal diversity-multiplexing tradeoff (DMT) in the high signal-to-noise-ratio (SNR) regime, and show that it equals $T$ times the DMT of the quasi-static channel. The converse is based on utilizing the …
In this paper, we first introduce the notion of channel leakage as the minimum mutual information between the channel input and channel output. As its name indicates, channel leakage quantifies the minimum information leakage to the malicious receiver. In a broad sense, it can be viewed as a dual concept of channel capacity, which characterizes the maximum information transmission to the targeted receiver. We obtain explicit formulas of channel leakage for the white Gaussian case, the colored Gaussian case, and the fading case. We then utilize this notion to investigate the fundamental limitat…
