We analyze the trade-off between the undetected error probability (i.e., the probability that the channel decoder outputs an erroneous message without detecting the error) and the total error probability in the short blocklength regime. We address the problem by developing two new finite blocklength achievability bounds, which we use to benchmark the performance of two coding schemes based on polar codes with outer cyclic redundancy check (CRC) codes—also referred to as CRC-aided (CA) polar codes. The first bound is obtained by considering an outer detection code, whereas the second bound relies on a threshold test applied to the generalized information density. Similarly, in the first CA polar code scheme, we reserve a fraction of the outer CRC parity bits for error detection, whereas in the second scheme, we apply a threshold test (specifically, Forney’s optimal rule) to the output of the successive cancellation list decoder. Numerical simulations performed on the binary-input AWGN channel reveal that, in the short-blocklength regime, the threshold-based approach is superior to the CRC-based approach, both in terms of bounds and performance of CA polar code schemes. We also consider the case of decoding with noisy channel-state information, which leads to a mismatched decoding setting. Our results illustrate that, differently from the previous case, in this scenario, the CRC-based approach outperforms the threshold-based approach, which is more sensitive to the mismatch.

This work addresses physical layer design aspects of coherent free-space optical downlinks from low-earth orbit satellites to ground. Achievable information rates are derived and assessed that include the availability of diversity, shaping, bit-metric decoding, repetition coding and automatic repeat request with maximum-ratio combining. A channel coding scheme is presented that approaches the theoretic limits within 1 dB. Extrinsic information transfer analysis for the free-space optical fading channel shows that a code design tailored to the additive white Gaussian noise channel is robust for fading channels with various parameters.