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arXiv2025

Low-Complexity Decoding for Low-Rate Block Codes of Short Length Based on Concatenated Coding Structure

To decode a short linear block code, ordered statics decoding (OSD) and/or the $A^*$ decoding are usually considered. Either OSD or the $A^*$ decoding utilizes the magnitudes of the received symbols to establish the most reliable and independent positions (MRIP) frame. A restricted searched space can be employed to achieve near-optimum decoding with reduced decoding complexity. For a low-rate code with large minimum distance, the restricted search space is still very huge. We propose to use concatenated coding to further restrict the search space by proposing an improved MRIP frame. The improved MRIP frame is founded according to magnitudes of log likelihood ratios (LLRs) obtained by the soft-in soft-out (SISO) decoder for the inner code. We focus on the construction and decoding of several $(n,k)$ = (128,36) binary linear block codes based on concatenated coding. We use the (128,36) extended BCH (eBCH) code as a benchmark for comparison. Simulation shows that there exist constructed concatenated codes which are much more efficient than the (128,36) eBCH code. Some other codes of length 128 or close to 128 are also constructed to demonstrate the efficiency of the proposed scheme.

cs.IT↗

arXiv2018

Output feedback stabilization for heat equations with sampled-data controls

In this paper, we build up an output feedback law to stabilize a sampled-data controlled heat equation (with a potential) in a bounded domain $\Omega$. The feedback law abides the following rules: First, we divide equally the time interval $[0,+\infty)$ into infinitely many disjoint time periods, and divide each time period into three disjoint subintervals. Second, for each time period, we observe a solution over an open subset of $\Omega$ in the first subinterval, take sample from outputs at one time point of the first subinterval, add a time-invariant output feedback control over another open subset of $\Omega$ in the second subinterval; let the equation evolve free in the last subinterval. Thus, the corresponding feedback control is of sampled-data. Our feedback law has the following advantages: the sampling period (which is the length of the above time period) can be arbitrarily taken; the feedback law has an explicit expression in terms of the sampling period; the behaviors of the norm of the feedback law, when the sampling period goes to zero or infinity, are clear. The construction of the feedback law is based on two kinds of approximate null-controllability for heat equations. One has time-invariant controls, while another has impulse controls. The studies of the aforementioned controllability with time-invariant controls need a new observability inequality for heat equations built up in the current work.

math.OC↗