Capacity on BMS Channels via Code Symmetry and Nesting

The past decade has seen notable advances in our understanding of structured error-correcting codes, particularly binary Reed--Muller (RM) codes. While initial breakthroughs were for erasure channels based on symmetry, extending these results to the binary symmetric channel (BSC) and other binary memoryless symmetric (BMS) channels required new tools and conditions. Recent work uses nesting to obtain multiple weakly correlated "looks" that imply capacity-achieving performance under bit-MAP and block-MAP decoding. This paper revisits and extends past approaches, aiming to simplify proofs, unify insights, and remove unnecessary conditions. By leveraging powerful results from the analysis of boolean functions, we derive recursive bounds using two or three looks at each stage. This gives bounds on the bit error probability that decay exponentially in the number of stages. For the BSC, we incorporate level-k inequalities and hypercontractive techniques to achieve the faster decay rate required for vanishing block error probability. The results are presented in a semitutorial style, providing both theoretical insights and practical implications for future research on structured codes.