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We explore the relationship between polar and RM codes and we describe acoding scheme which improves upon the performance of the standard polar code atpractical block lengths. Our starting point is the experimental observationthat RM codes have a smaller error probability than polar codes under MAPdecoding. This motivates us to introduce a family of codes that "interpolates"between RM and polar codes, call this family ${\mathcal C}_{\rm inter} =\{C_{\alpha} : \alpha \in [0, 1]\}$, where $C_{\alpha} \big |_{\alpha = 1}$ isthe original polar code, and $C_{\alpha} \big |_{\alpha = 0}$ is an RM code.Based on numerical observations, we remark that the error probability under MAPdecoding is an increasing function of $\alpha$. MAP decoding has in generalexponential complexity, but empirically the performance of polar codes atfinite block lengths is boosted by moving along the family ${\mathcal C}_{\rminter}$ even under low-complexity decoding schemes such as, for instance,belief propagation or successive cancellation list decoder. We demonstrate theperformance gain via numerical simulations for transmission over the erasurechannel as well as the Gaussian channel.

From Polar to Reed-Muller Codes: a Technique to Improve the Finite-Length Performance