Random Linear Streaming Codes in the Finite Memory Length and Decoding Deadline Regime—Part I: Exact Analysis

Abstract

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Streaming codes</i> take a string of source symbols as input and output a string of coded symbols in real time, which eliminate the queueing delay of traditional <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">block codes</i> and are thus especially appealing for delay sensitive applications. Existing works on streaming code performance either focused on the asymptotic error-exponent analyses, or on the optimal code construction under <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">deterministic adversarial channel models</i> . In contrast, this work analyzes the exact error probability of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">random linear streaming codes</i> (RLSCs) in the large field size regime over the stochastic i.i.d. symbol erasure channel model. A closed-form expression of the error probability of large-field-size RLSCs is derived under, simultaneously, the finite memory length and decoding deadline constraints. The result is then used to examine the intricate tradeoff between memory length (complexity), decoding deadline (delay), code rate (throughput), and error probability (reliability). Numerical evaluation shows that under the same code rate and error probability requirements, the end-to-end delay of RLSCs is 40–48% of that of the optimal block codes (i.e., MDS codes). This implies that switching from block codes to streaming codes not only eliminates the queueing delay completely (which accounts for the initial 50% of the delay reduction) but also improves the reliability (which accounts for the additional 2–10% delay reduction).