Interleaved Polar (I-Polar) Codes

By inserting interleavers between intermediate stages of the polar encoder, a new class of polar codes, termed interleaved polar (i-polar) codes, is proposed. By the uniform interleaver assumption, we derive the weight enumerating function (WEF) and the input-output weight enumerating function (IOWEF) averaged over the ensemble of i-polar codes. The average WEF can be used to calculate the upper bound on the average block error rate (BLER) of a code selected at random from the ensemble of i-polar codes. Also, we propose a concatenated coding scheme that employs <inline-formula> <tex-math notation="LaTeX">$P$ </tex-math></inline-formula> high rate codes as the outer code and <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> i-polar codes as the inner code with an interleaver in between. The average WEF of the concatenated code is derived based on the uniform interleaver assumption. Simulation results show that BLER upper bounds can well predict BLER performance levels of the concatenated codes. The results show that the performance of the proposed concatenated code with <inline-formula> <tex-math notation="LaTeX">$P=Q=2$ </tex-math></inline-formula> is better than that of the CRC-aided i-polar code with <inline-formula> <tex-math notation="LaTeX">$P=Q=1$ </tex-math></inline-formula> of the same length and code rate at high signal-to-noise ratios (SNRs). Moreover, the proposed concatenated code allows multiple decoders to operate in parallel, which can reduce the decoding latency and hence is suitable for ultra-reliable low-latency communications (URLLC).