The joint detection uses Kalman filtering (KF) to estimate the prior probability of control outputs to assist channel decoding. In this paper, we regard the joint detection as maximum a posteriori (MAP) decoding and derive the lower and upper bounds based on the pairwise error probability considering system interference, quantization interval, and weight distribution. We first derive the limiting bounds as the signal-to-noise ratio (SNR) goes to infinity and the system interference goes to zero. Then, we construct an infinite-state Markov chain to describe the consecutive packet losses of the control systems to derive the MAP bounds. Finally, the MAP bounds are approximated as the bounds of the transition probability from the state with no packet loss to the state with consecutive single packet loss. The simulation results show that the MAP performance of $\left(64,16\right)$ polar code and 16-bit CRC coincides with the limiting upper bound as the SNR increases and has $3.0$dB performance gain compared with the normal approximation of the finite block rate at block error rate $10^{-3}$.

Minimum weight distribution (MWD) is an important metric to calculate the first term of union bound called minimum weight union bound (MWUB). In this paper, we first prove the maximum likelihood (ML) performance approaches MWUB as signal-to-noise ratio (SNR) goes to infinity and provide the deviation when MWD and SNR are given. Then, we propose a nested reliability sequence, namely MWD sequence, to construct polar codes independently of channel information. In the sequence, synthetic channels are sorted by partial MWD which is used to evaluate the influence of information bit on MWD and we prove the MWD sequence is the optimum sequence evaluated by MWUB for polar codes obeying partial order. Finally, we introduce an entropy constraint to establish a relationship between list size and MWUB and propose a heuristic construction method named entropy constraint bit-swapping (ECBS) algorithm, where we initialize information set by the MWD sequence and gradually swap information bit and frozen bit to satisfy the entropy constraint. The simulation results show the MWD sequence is more suitable for constructing polar codes with short code length than the polar sequence in 5G and the ECBS algorithm can improve MWD to show better performance as list size increases.

We propose a rate-filling method as a framework to study fast construction of bit-interleaved polar-coded modulation (BIPCM). In particular, the proposed method makes full use of the nesting structure of polar codes. We show that BIPCM can be constructed by appropriately allocating the rates of each polar subcode whose information set is easily extracted by a predetermined index sequence, e.g., the Polar sequence in the 5G standard. By this means, the reliability calculation and ranking operation of bit channels can be discarded, which is a desired feature for low-complex construction. Simulation results show the proposed fast construction method performs independently of the actual channel condition and is robust to diverse modulation and coding schemes in the 5G standard.