Reed-Muller (RM) codes are one of the oldest families of codes. Recently, a recursive projection aggregation (RPA) decoder has been proposed, which achieves a performance that is close to the maximum likelihood decoder for short-length RM codes. One of its main drawbacks, however, is the large amount of computations needed. In this paper, we devise a new algorithm to lower the computational budget while keeping a performance close to that of the RPA decoder. The proposed approach consists of multiple sparse RPAs that are generated by performing only a selection of projections in each sparsified decoder. In the end, a cyclic redundancy check (CRC) is used to decide between output codewords. Simulation results show that our proposed approach reduces the RPA decoder's computations up to 80% with negligible performance loss.
We consider the problem of coded distributed computing using polar codes. The average execution time of a coded computing system is related to the error probability for transmission over the binary erasure channel in recent work by Soleymani, Jamali and Mahdavifar, where the performance of binary linear codes is investigated. In this paper, we focus on polar codes and unveil a connection between the average execution time and the scaling exponent µ of the family of codes. The scaling exponent has emerged as a central object in the finitelength characterization of polar codes, and it captures the speed of convergence to capacity. In particular, we show that (i) the gap between the normalized average execution time of polar codes and that of optimal MDS codes is O(n −1/µ ), and (ii) this upper bound can be improved to roughly O(n −1/2 ) by considering polar codes with large kernels. We conjecture that these bounds could be improved to O(n −2/µ ) and O(n −1 ), respectively, and provide a heuristic argument as well as numerical evidence supporting this view.
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