Preserving the Minimum Distance of Polar-Like Codes while Increasing the Information Length

Abstract

Reed Muller (RM) codes are known for their good minimum distance. One can use their structure to construct polar-like codes with good distance properties by choosing the information set as the rows of the polarization matrix with the highest Hamming weight, instead of the most reliable synthetic channels. However, the information length options of RM codes are quite limited due to their specific structure. In this work, we present sufficient conditions to increase the information length by at least one bit for some underlying RM codes and in order to obtain pre-transformed polar-like codes with the same minimum distance than lower rate codes. Moreover, our findings are combined with the method presented in [1] to further reduce the number of minimum weight codewords. Numerical results show that the designed codes perform close to the meta-converse bound at short blocklengths and better than the polarized adjusted convolutional polar codes with the same parameters.

Dynamic Frozen-Function Design for Reed-Muller Codes With Automorphism-Based Decoding

“…Next, we prove that the dynamic freezing constraint matrix V induced by (9), compliant with the concept of singleargument dynamic frozen functions [8], is stable after applying any permutation π ∈ BLTA(n − 1, 1) ∩ PL.…”

“…V transcribes the bits in F , frozen at 0 (with a single 1 in the row) or data-dependent (several 1s in one row). Properly chosen pre-transformation matrices may reduce the number of low-weight codewords, enhancing the distance properties in the finite length regime [9], [10]. Reed-Muller codes are often adopted as underlying codes for pre-transformed polar codes at short block lengths [6], [8], [9].…”

“…In the former case, we have u i = u j = 0, in the latter case, the bit i is a dynamic frozen bit with u i = u j ∈ {0, 1}. Hence, in most cases, the matrix V, that transcribes the bits in F as induced by the constraint (9), can be row reduced. The maximum number of dynamic frozen bits D is given by…”

“…On the other hand, recently introduced Polarizationadjusted-convolutional (PAC) codes [6] perform very close to the second-order rate approximation of the binary-input additive white Gaussian noise (BI-AWGN) in short blocklength regime [7] under SCL decoding with large list sizes, i.e., L ≥ 128. In [8] and [9], the concept of single-argument dynamic frozen functions was introduced, where any dynamic frozen bit is dependent on only one previous information bit. Also, it has been shown that properly chosen dynamic frozen bits reduce the number of minimum weight codewords or increase the minimum distance.…”

“…With properly chosen pre-transformation matrices, pre-transformed polar codes were proven to enhance the distance spectrum of the code without reducing the minimum distance of the underlying code [10]. Given their distance properties, Reed-Muller codes are often chosen as underlying codes of pretransformed polar codes for short block lengths [6], [8], [9].…”