Linear error correcting codes with anytime reliability

Abstract

We consider rate R = k/n causal linear codes that map a sequence of k-dimensional binary vectors {b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t=0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> to a sequence of n-dimensional binary vectors {c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t=0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sup> , such that each c <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> is a function of {b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">τ=0</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> . Such a code is called anytime reliable, for a particular binary-input memoryless channel, if at each time instant t, and for all delays d ≥ d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> , the probability of error P(b̂ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-d/t</sub> ≠b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-d</sub> ) decays exponentially in d, i.e., P(b̂ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-d/t</sub> ≠b <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t-d</sub> ) ≤ η2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-βnd</sup> , for some β >; 0. Anytime reliable codes are useful in interactive communication problems and, in particular, can be used to stabilize unstable plants across noisy channels. Schulman proved the existence of such codes which, due to their structure, he called tree codes in [1]; however, to date, no explicit constructions and tractable decoding algorithms have been devised. In this paper, we show the existence of anytime reliable “linear” codes with “high probability”, i.e., suitably chosen random linear causal codes are anytime reliable with high probability. The key is to consider time-invariant codes (i.e., ones with Toeplitz generator and parity check matrices) which obviates the need to union bound over all times. For the binary erasure channel we give a simple ML decoding algorithm whose average complexity is constant per time instant and for which the probability that complexity at a given time t exceeds KC <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> decays exponentially in C. We show the efficacy of the method by simulating the stabilization of an unstable plant across a BEC, and remark on the tradeoffs between the utilization of the communication resources and the control performance.