RPA Decoding with Few Subspace Projections for Correcting Structured Error Patterns in RM Codes

Abstract

The Recursive Projection Aggregation (RPA) decoder for Reed-Muller codes uses, in its default implementation, all one-dimensional subspaces at each level of recursion. We restrict our attention to the case where the number of errors is $t<d_{\mathrm{min}}/2$. It is known that if the number of errors is $t<d_{\mathrm{min}}/2$, then $2t+1$ subspaces at each level of recursion are sufficient to correct these errors. We show that, for structured error patterns, the number of subspaces required for correction can be reduced further by appropriately selecting the subspaces used for projection. In particular, when errors occur as a single burst of length $t < d_{\mathrm{min}}/2$, three subspaces at each level of recursion suffice for correction. We obtain similar results for multiple bursts and other structured error patterns as well.