Partial Orders in Rate-Matched Polar Codes

TL;DR

This paper establishes and leverages partial orders in rate-matched polar codes for BEC and BMSC, improving reliable sequence construction and demonstrating persistence of these orders after rate matching.

Contribution

It introduces the concept of partial orders in rate-matched polar codes and demonstrates their persistence and utility in improving reliable sequences.

Findings

POs are established for BEC and BMSC under rate matching.

POs persist after block rate matching in BEC.

Using POs improves the PW sequence from b2-expansion.

Abstract

In this paper, we establish the partial order (POs) for both the binary erasure channel (BEC) and the binary memoryless symmetric channel (BMSC) under any block rate-matched polar codes. Firstly, we define the POs in the sense of rate-matched polar codes as a sequential block version. Furthermore, we demonstrate the persistence of POs after block rate matching in the BEC. Finally, leveraging the existing POs in the BEC, we obtain more POs in the BMSC under block rate matching. Simulations show that the PW sequence constructed from \beta-expansion can be improved by the tool of POs. Actually, any fixed reliable sequence in the mother polar codes can be improved by POs for rate matching.