Related papers: Weight Structure of Low/High-Rate Polar Codes and …
In this article, we provide a complete characterization of codewords in polar codes with weights less than twice the minimum distance, using the group action of the lower triangular affine (LTA) group. We derive a closed-form formula for…
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 $w_{\min}$…
Polar codes can be viewed as decreasing monomial codes, revealing a rich algebraic structure governed by the lower-triangular affine (LTA) group. We develop a general framework to compute the Hamming weight of codewords generated by sums 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,…
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…
Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of linear codes has many applications such as on the type II wire-tap channel, dealing with…
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…
Linear codes with few weights have applications in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs. This paper first generalizes the method of…
Currently known secondary construction techniques for linear codes mainly include puncturing, shortening, and extending. In this paper, we propose a novel method for the secondary construction of linear codes based on their weight…
Linear codes with few weights have significant applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. There are a number of methods to construct linear codes, one of which is based on…
The code spectrum of polar codes is crucial to the performance of polar codes. Based on the lower-triangular affine group (LTA) of decreasing monomial codes and the one-variable descendance (ovd) relation, we define a new subgroup of LTA…
Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attain many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both…
Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many…
For the past decades, linear codes with few weights have been widely studied, since they have applications in space communications, data storage and cryptography. In this paper, a class of binary linear codes is constructed and their weight…
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…
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…
Linear codes have diverse applications in secret sharing schemes, secure two-party computation, association schemes, strongly regular graphs, authentication codes and communication. There are a large number of linear codes with few weights…
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…
Linear codes with few weights have applications in authentication codes, secrete sharing schemes, association schemes, consumer electronics and data storage system. In this paper, several classes of linear codes with two or three weights…
We introduce a formula for determining the number of codewords of weight 2 in cyclic codes and provide results related to the count of codewords with weight 3. Additionally, we establish a recursive relationship for binary cyclic codes that…