Linear error correcting codes with anytime reliability

We consider rate R = k/n causal linear codes that map a sequence of k-dimensional binary vectors {b_t}_t=0^∞ to a sequence of n-dimensional binary vectors {c_t}_t=0^∞, such that each c_t is a function of {b_τ}_τ=0^t. 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_o, the probability of error P(b̂_t-d/t≠b_t-d) decays exponentially in d, i.e., P(b̂_t-d/t≠b_t-d) ≤ η2^-βnd, 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 KC³ 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.