Weight Distribution of Polar Codes

This repository contains two MATLAB scripts for weight enumeration and computing the partial or complete weight distribution (also known as the weight spectrum) of polar codes.

The scripts are organized into two folders:

1. Closed-form Formulas:
   This folder contains a script that uses closed-form formulas to compute the partial weight distribution of polar codes for weights less than twice the minimum distance $(< 2 \cdot w_{\min})$, leveraging group actions on maximum-degree monomials. Based on the theoretical results in [1–3] and further developed in [4], this method is efficient and applicable to short, medium, and long polar codes.

2. Recursive Weight Enumerator (WEF):
   This folder provides MATLAB scripts to compute the Weight Enumerator Function (WEF) of polar codes using recursive enumeration, yielding the complete weight distribution for short codes. The algorithm is a variation of the approach in [4], with detailed explanation and an illustrative example provided in [5].

References

[1] V. -F. Drăgoi, M. Rowshan and J. Yuan, "On the Closed-Form Weight Enumeration of Polar Codes: 1.5d -Weight Codewords," in IEEE Transactions on Communications, vol. 72, no. 10, pp. 5972–5987, Oct. 2024, doi: 10.1109/TCOMM.2024.3394749.

[2] M. Rowshan, V. Drăgoi and J. Yuan, "Weight Structure of Low/High-Rate Polar Codes and Its Applications," in 2024 IEEE International Symposium on Information Theory (ISIT), Athens, Greece, 2024, pp. 2945–2950, doi: 10.1109/ISIT57864.2024.10619618.

[3] V.-F. Dragoi and M. Rowshan, "On weight enumeration and structure characterization of polar codes via group actions," arXiv:2504.19544v2, 2025. [Online]. Available: https://arxiv.org/abs/2504.19544

[4] H. Yao, A. Fazeli and A. Vardy, "A Deterministic Algorithm for Computing the Weight Distribution of Polar Code," in IEEE Transactions on Information Theory, vol. 70, no. 5, pp. 3175–3189, May 2024, doi: 10.1109/TIT.2023.3243191.

[5] M. Rowshan, V.-F. Dragoi, "A Tutorial on Weight Structure of Polar Codes," submitted to the Special Issue on Error-Correcting Codes of IEEE BITS magazine.