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IEEE Transactions on Communications

On the Closed-Form Weight Enumeration of Polar Codes: 1.5 d -Weight Codewords

Vlad Drăgoi, Mohammad Rowshan, Jinhong Yuan

Published May 6, 2024• 18 citations
Publisher

Abstract

The weight distribution of an error correction code is a critical determinant of its error-correcting performance. In the case of polar codes, the minimum weight wmin(equal to the minimum distanced) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight 1.5wmin. The key insight for code design is that the enumeration of codewords with weight wmin and 1.5wminrelies on the set of maximum degree monomials. This set corresponds to the indices of minimum weight rows of the polar transformGNbelonging to the information setI. Consequently, reducing the cardinality of this set can lead to a reduction of the number of codewords in both weight categories.

DOI
10.1109/tcomm.2024.3394749

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The weight distribution of an error correction code is a critical determinant of its error-correcting performance. In the case of polar codes, the minimum weight wmin(equal to the minimum distanced) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight 1.5wmin. The key insight for code design is that the enumeration of codewords with weight wmin and 1.5wminrelies on the set of maximum degree monomials. This set corresponds to the indices of minimum weight rows of the polar transformGNbelonging to the information setI. Consequently, reducing the cardinality of this set can lead to a reduction of the number of codewords in both weight categories.
The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight $\wm$ (which is equal to minimum distance ddd) is the only weight for which an explicit enumerator formula is currently available. Having closed-form weight enumerators for polar codewords with weights greater than the minimum weight not only simplifies the enumeration process but also provides valuable insights towards constructing better polar-like codes. In this paper, we contribute towards understanding the algebraic structure underlying higher weights by analyzing Minkowski sums of orbits. Our approach builds upon the lower triangular affine (LTA) group of decreasing monomial codes. Specifically, we propose a closed-form expression for the enumeration of codewords with weight $1.5\wm$. Our simulations demonstrate the potential for extending this method to higher weights.
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.
On the Weight Distribution of Weights Less than 2wmin⁡2w_{\min}2wmin​ in Polar Codes
The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than 2wmin⁡2w_{\min}2wmin​, where wmin⁡w_{\min}wmin​ represents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than 2wmin⁡2w_{\min}2wmin​. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.
The number of low-weight codewords is critical to the performance of error-correcting codes. In 1970, Kasami and Tokura characterized the codewords of Reed-Muller (RM) codes whose weights are less than 2wmin, wherewminrepresents the minimum weight. In this paper, we extend their results to decreasing polar codes. We present the closed-form expressions for the number of codewords in decreasing polar codes with weights less than 2wmin. Moreover, the proposed enumeration algorithm runs in polynomial time with respect to the code length.
The minimum Hamming distance of a linear block code is the smallest number of bit changes required to transform one valid codeword into another. The code’s minimum distance determines the code’s error-correcting capabilities. Furthermore, The number of minimum weight codewords, a.k.a. error coefficient, gives a good comparative measure for the block error rate (BLER) of linear block codes with identical minimum distance, in particular at a high SNR regime under maximum likelihood (ML) decoding. A code with a smaller error coefficient would give a lower BLER. Unlike polar codes, a closed-form expression for the enumeration of the error coefficient of polarization-adjusted convolutional (PAC) codes is yet unknown. As PAC codes are convolutionally pre-transformed polar codes, we study the impact of pre-transformation on polar codes in terms of minimum Hamming distance and error coefficient by partitioning the codewords into cosets. We show that the minimum distance of PAC codes does not decrease; however, the pre-transformation may reduce the error coefficient depending on the choice of convolutional polynomial. We recognize the properties of the cosets where pre-transformation is ineffective in decreasing the error coefficient, giving a lower bound for the error coefficient. Then, we propose a low-complexity enumeration method that determines the number of minimum weight codewords of PAC codes relying on the error coefficient of polar codes. That is, given the error coefficient Awmin of polar codes, we determine the reduction X in the error coefficient due to convolutional pre-transformation in PAC coding and subtract it from the error coefficient of polar codes, Awmin - X. Furthermore, we numerically analyze the tightness of the lower bound and the impact of the choice of the convolutional polynomial on the error coefficient based on the sub-patterns in the polynomial’s coefficients. Eventually, we show how we can further reduce the error coefficient in the cosets.
Polar codes are usually constructed by ranking synthetic bit-channels according to reliability, which guarantees capacity-achieving behavior but can yield poor low-weight spectra at short and moderate lengths. Recent algebraic results express the contribution of individual bit-channels to the multiplicities of minimum and near-minimum weight codewords in closed form. In this work we combine these insights into a mixed (reliability--weight) bit-channel ordering. We define a per-bit cost whose distance term is derived from orbit enumeration of minimum-weight codewords and scaled by a Bhattacharyya-type factor, and show that the resulting mixed construction minimises a truncated SC/ML union-bound surrogate within a class of decreasing monomial codes. We relate the mixed metric to error events in SCL decoding via a pruning/ML decomposition, and prove that mixed designs act as local perturbations of reliability-based constructions whose asymptotic impact vanishes as code-length approaches infinity. Numerical results for short and moderate lengths on BPSK-AWGN, implemented via Gaussian approximation and closed-form weight contributions, illustrate the trade-off between pure reliability-based and mixed constructions in terms of minimum distance, multiplicity, and union-bound approximations. All proofs are deferred to the appendices.