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arXiv:2601.10262 · IEEE International Symposium on Information Theory (ISIT)

Algebraic Properties of PAC Codes

Vlad Drăgoi, Mohammad Rowshan

Published Jun 1, 2026• 1 citation
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Abstract

We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code.

DOI
10.1109/isit62367.2026.11653640

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We analyze polarization-adjusted convolutional codes using the algebraic representation of polar and Reed-Muller codes. We define a large class of codes, called generalized polynomial polar codes which include PAC codes and Reverse PAC codes. We derive structural properties of generalized polynomial polar codes, such as duality, minimum distance. We also deduce some structural limits in terms of number of minimum weight codewords, and dimension of monomial sub-code.
Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attains many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a nontrivial task and is channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised for a binary-input symmetric channel always form a decreasing set. We call such codes decreasing monomial codes. The fact that polar codes are decreasing monomial codes turns out to have rather deep consequences on their structure. Indeed, we show that decreasing monomial codes have a very large permutation group by proving that it contains a group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from evaluations of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.
Polar codes and Reed-Muller codes belong to a family of codes called monomial codes. In this work, we study pre-transformed monomial codes, which cover several constructions including parity-check (PC) codes and PAC codes. We show that any pre-transformed monomial code can be transformed into a parity-check monomial code with the same codewords, and give an explicit algorithm for this transformation. We further prove that for certain monomial codes, the minimum weight is invariant under pre-transformation, but specific pre-transformation matrices can be constructed to reduce the number of minimum-weight codewords. These results offer theoretical support for the success of various heuristics, e.g., PAC codes attain dispersion bound, and provide guidance for designing short codes.
Convolutional precoding in polarization-adjusted convolutional (PAC) codes is a recently introduced variant of polar codes. It has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective, this reduction has limitations. Capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper studies reverse precoding that can effectively reduce minimum-weight codewords more than in PAC codes. We propose a look-ahead list decoding for the reverse PAC codes, which has the same order of complexity as list decoding in PAC codes. Through numerical analysis, we demonstrate a notable reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate, in particular at high code rates.
Convolutional precoding in polarization-adjusted convolutional (PAC) codes, a recently introduced variant of polar codes, has demonstrated an effective reduction in the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This reduction has the potential to significantly improve the error correction performance. From a codeword formation perspective in cosets, this reduction has a limitation in the PAC coding which depends on the rows of the generator matrix involved in the formation of codewords. To overcome this limitation, capitalizing on the understanding of the decomposition of minimum-weight codewords, this paper introduces a novel precoding scheme that strategically disrupts the formation of a majority of minimum-weight codewords. This scheme significantly enhances the error coefficient without compromising the minimum distance of the code. Through numerical analysis, we demonstrate a noteworthy reduction in error coefficients compared to PAC codes and polar codes, resulting in a remarkable improvement in the block error rate of short codes.
In polarization-adjusted convolutional (PAC) codes, a convolutional transform is employed as a precoding stage before the polar transform. It turned out that the precoding stage improves the weight distribution of polar codes according. That could be the main reason for the superiority of error correction performance of PAC codes relative to polar codes. In this work, we explicitly show why the convolutional precoding reduces the number of minimum-weight codewords. Furthermore, we show where the precoding stage is not effective in the improvement of weight distribution. Eventually, to evaluate the FER gain obtainable from reducing the number of min-weight codewords, we assess the performance and the number of minimum-weight codewords resulted from an irregular convolutional precoding process.
Polarization-adjusted convolutional (PAC) codes have better error-correcting performance than polar codes mostly because of the improved weight distribution brought by the convolutional pre-transformation. In this paper, we propose the parity check PAC (PC-PAC) codes to further improve error-correcting performance of PAC codes. The design principle is to establish parity check functions between bits with distinct row weights, such that information bits of lower reliability are re-protected by the PC relation. Moreover, an algorithm to select which bits to be involved in parity-check functions is also proposed to make sure that the constructed codes have fewer minimum-weight codewords. Simulation results show that the proposed PC-PAC codes can achieve nearly 0.2dB gain over PAC codes at frame error rate (FER) about 10-3 codes.