On the formation of min-weight codewords of polar/PAC codes and its applications

Mohammad Rowshan, Son Hoang Dau, Emanuele Viterbo

Research output: Contribution to journalArticleResearchpeer-review


Minimum weight codewords play a crucial role in the error correction performance of a linear block code. In this work, we establish an explicit construction for these codewords of polar codes as a sum of the generator matrix rows, which can then be used as a foundation for two applications. In the first application, we obtain a lower bound for the number of minimum-weight codewords (a.k.a. the error coefficient), which matches the exact number established previously in the literature. In the second application, we derive a novel method that modifies the information set (a.k.a. rate profile) of polar codes and PAC codes in order to reduce the error coefficient, hence improving their performance. More specifically, by analyzing the structure of minimum-weight codewords of polar codes (as special sums of the rows in the polar transform matrix), we can identify rows (corresponding to <italic>information</italic> bits) that contribute the most to the formation of such codewords and then replace them with other rows (corresponding to <italic>frozen</italic> bits) that bring in few minimum-weight codewords. A similar process can also be applied to PAC codes. Our approach deviates from the traditional constructions of polar codes, which mostly focus on the reliability of the sub-channels, by taking into account another important factor - the weight distribution. Extensive numerical results show that the modified codes outperform PAC codes and CRC-Polar codes at the practical block error rate of 10-2-10-3.

Original languageEnglish
Pages (from-to)7627-7649
Number of pages23
JournalIEEE Transactions on Information Theory
Issue number12
Publication statusPublished - Dec 2023


  • code construction
  • Codes
  • list decoding
  • Maximum likelihood decoding
  • minimum Hamming distance
  • PAC Codes
  • Picture archiving and communication systems
  • Polar codes
  • polar codes
  • Polarization-adjusted convolutional codes
  • rate profile
  • Reliability
  • Signal to noise ratio
  • Transforms
  • weight distribution

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