Successive Cancellation Ordered Search Decoding of Modified $\boldsymbol{G}_N$-Coset Codes

A tree search algorithm called successive cancellation ordered search (SCOS) is proposed for $\boldsymbol{G}_N$-coset codes that implements maximum-likelihood (ML) decoding with an adaptive complexity for transmission over binary-input AWGN channels. Unlike bit-flip decoders, no outer code is needed to terminate decoding; therefore, SCOS also applies to $\boldsymbol{G}_N$-coset codes modified with dynamic frozen bits. The average complexity is close to that of successive cancellation (SC) decoding at practical frame error rates (FERs) for codes with wide ranges of rate and lengths up to $512$ bits, which perform within $0.25$ dB or less from the random coding union bound and outperform Reed--Muller codes under ML decoding by up to $0.5$ dB. Simulations illustrate simultaneous gains for SCOS over SC-Fano, SC stack (SCS) and SC list (SCL) decoding in FER and the average complexity at various SNR regimes. SCOS is further extended by forcing it to look for candidates satisfying a threshold on the likelihood, thereby outperforming basic SCOS under complexity constraints. The modified SCOS enables strong error-detection capability without the need for an outer code. In particular, the $(128, 64)$ PAC code under modified SCOS provides gains in overall and undetected FER compared to CRC-aided polar codes under SCL/dynamic SC flip decoding at high SNR.