On the Average Weight Spectrum of Pre-Transformed Rate-Compatible Polar Codes

The weight spectrum plays a crucial role in the performance of error-correcting codes. Pre-transformation with an upper-triangular matrix improves the weight spectrum of polar codes while retaining polarization. However, a theoretical analysis to quantify the improvement for pre-transformed rate-compatible polar codes is missing. In this paper, we calculate the average spectrum of random upper-triangular pre-transformed shortened and punctured polar codes. Our approach tran-scends the limitations imposed by partial ordering and specific rate matching patterns. A key feature of our approach is its polynomial complexity in relation to the code length, making it computationally feasible. Simulation results affirm that our findings provide an accurate approximation on the performance of pre-transformed rate-compatible polar codes.