Related papers: On the Weight Spectrum Improvement of Pre-transfor…
Polar codes are the first class of channel codes achieving the symmetric capacity of the binary-input discrete memoryless channels with efficient encoding and decoding algorithms. But the weight spectrum of Polar codes is relatively poor…
In this paper, we prove that any pre-transformation with an upper-triangular matrix (including cyclic redundancy check (CRC), parity-check (PC) and convolution code (CC) matrix) does not reduce the code minimum distance and an properly…
Pre-transformed polar codes (PTPCs) form a class of codes that perform close to the finite-length capacity bounds. The minimum distance and the number of minimum weight codewords are two decisive properties for their performance. In this…
CRC-Polar codes under SC list decoding are well-regarded for their competitive error performance. This paper examines these codes by focusing on minimum weight codewords, breaking them down into the rows of the polar transform. Inspired by…
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…
Polarization-adjusted convolutional (PAC) codes are special concatenated codes in which we employ a one-to-one convolutional transform as a precoding step before the polar transform. In this scheme, the polar transform (as a mapper) and the…
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…
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 are constructed based on the reliability of sub-channels resulting from the polarization effect. However, this information-theoretic construction approach leads to a poor weight distribution. To address this issue,…
The weight spectrum plays a crucial role in the performance of error-correcting codes. Despite substantial theoretical exploration of polar codes with mother code length, a framework for the weight spectrum of rate-compatible 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 $2w_{\min}$, where $w_{\min}$…
In this paper, we present a deterministic algorithm to count the low-weight codewords of punctured and shortened pure and pre-transformed polar codes. The method first evaluates the weight properties of punctured/shortened polar cosets.…
Deep polar codes, employing multi-layered polar kernel pre-transforms in series, are recently introduced variants of pre-transformed polar codes. These codes have demonstrated the ability to reduce the number of minimum weight codewords,…
Convolutional precoding in polarization-adjusted convolutional (PAC) codes can reduce the number of minimum weight codewords (a.k.a error coefficient) of polar codes. This can result in improving the error correction performance of (near)…
Reed-Muller (RM) codes exhibit good performance under maximum-likelihood (ML) decoding due to their highly-symmetric structure. In this paper, we explore the question of whether the code symmetry of RM codes can also be exploited to achieve…
We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM…
Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are…
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…
Polar codes have promising error-correction capabilities. Yet, decoding polar codes is often challenging, particularly with large blocks, with recently proposed decoders based on list-decoding or neural-decoding. The former applies multiple…
Row-merged polar codes are a family of pre-transformed polar codes (PTPCs) with little precoding overhead. Providing an improved distance spectrum over plain polar codes, they are capable to perform close to the finite-length capacity…