Abstract
A novel method to recursively construct a set of precoded polar codes of various rates and short-to-moderate lengths is presented. The proposed code design method minimizes the successive cancellation (SC) decoding error probability estimate under three constraints. The first constraint is the minimum distance requirement to improve the maximum-likelihood (ML) performance of the resulting code and therefore the performance under the SC list (SCL) decoding. The other two constraints introduce preselected supercode and subcode, where the supercode ensures fast computation of the minimum distance and the subcode ensures reduction of the search space size. The supercode is given by the Plotkin sum of shorter codes, which are nested to simplify computation of low-weight codewords. These low-weight codewords are needed to satisfy the minimum distance constraint. The simulation results indicate that the proposed precoded polar codes of lengths 128 and 256 provide a better frame error rate (FER) than polar codes with CRC and e-BCH polar subcodes under the SCL decoding algorithm with the list size <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$8-128$ </tex-math></inline-formula> .
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