Weight Codewords Research Articles

Weights of Codewords in Random Binary Linear Codes and Their Correlation Properties

Weights of Codewords in Random Binary Linear Codes and Their Correlation Properties

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

On the Closed-Form Weight Enumeration of Polar Codes: 1.5<i>d</i> -Weight Codewords

New distance bounds for quasi-cyclic codes

AbstractWe consider the minimum weight of codewords in a quasi-cyclic code and characterize the estimate in its most general setup using their concatenated structure. The new bound we derive generalizes the Jensen and Güneri–Özbudak bounds and it holds for the more general class of multilevel concatenated codes.

Small weight codewords of projective geometric codes II

Small weight codewords of projective geometric codes II

Identifying codewords in general Reed-Muller codes and determining their weights

&lt;abstract&gt;&lt;p&gt;Determining the weight distribution of all Reed-Muller codes is a huge and exciting problem that has been around since the sixties. Some progress has been made very recently, but we are still far from a solution. In this paper, we addressed the subproblem of determining as many codeword weights as possible in Reed-Muller codes of any lengths and any orders, which is decisive for determining their weight spectra (i.e., the lists of all possible weights in these codes). New approaches seem necessary for both the main problem and the subproblem. We first studied the difficulties and the limits of the approach, which consisted of using the usual primary and secondary constructions of Boolean functions for the purpose of determining as many weights as possible in Reed-Muller codes. We then introduced a way, different from the usual constructions, to generate Boolean functions in $ n $ variables having an algebraic degree bounded from above, without any restriction on $ n $, and whose Hamming weights can be determined. This provided weights in Reed-Muller codes of any lengths $ 2^n $ and any orders, allowing us to reach potentially new values in the weight spectra of Reed-Muller codes (as we illustrate with all Reed-Muller codes of lengths up to $ 2^{21} $), with the related codewords being given with their supports and their algebraic normal forms being mathematically derived.&lt;/p&gt;&lt;/abstract&gt;

On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes

On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes

Block designs from self-dual codes obtained from Paley designs and Paley graphs

In 2002, P. Gaborit introduced two constructions of self-dual codes using quadratic residues, so-called pure and bordered construction, as a generalization of the Pless symmetry codes. In this paper, we further study conditions under which the pure and the bordered construction using Paley designs and Paley graphs yield self-dual codes. Special attention is given to the binary and ternary codes. Further, we construct \(t\)-designs from supports of the codewords of a particular weight in the binary and ternary codes obtained.

Some combinatorial identities

In this paper, it is shown that a positive integer [Formula: see text] can be written as [Formula: see text]. It is also proved that the two expressions for number of codewords of weight [Formula: see text], obtained in Theorem 1[P. Kumar, M. Sangwan and S. K. Arora, The weight distributions of some irreducible cyclic codes of length [Formula: see text] and [Formula: see text], Adv. Math. Commun. 9 (2015) 277–289] and in Theorem 1 Case (i)[G. Vega, The weight distribution for any irreducible cyclic code of length [Formula: see text], Appl. Algebra Eng. Commun. Comput. 29(4) (2018) 363–370] are same.

New examples of self-dual near-extremal ternary codes of length 48 derived from 2-[formula omitted] designs

In a recent paper (Araya and Harada, 2023), Araya and Harada gave examples of self-dual near-extremal ternary codes of length 48 for 145 distinct values of the number A12 of codewords of minimum weight 12, and raised the question about the existence of codes for other values of A12. In this note, we use symmetric 2-(47,23,11) designs with an automorphism group of order 6 to construct self-dual near-extremal ternary codes of length 48 for 150 new values of A12.

On the codewords of generalized Reed-Muller codes reaching the fourth weight

Recently, the fourth weight of generalized Reed-Muller codes Rq(a(q−1)+b,m) have been characterized for 3≤b<q+43 and 0≤a≤m−3. In this paper, we study the codewords reaching the fourth weight of generalized Reed-Muller codes. We will determine the fourth weight codewords of generalized Reed-Muller codes of order r=a(q−1)+b with 3≤b<q+43 and 0≤a≤m−3. Also, we characterize the number of fourth weight codewords of Rq(b,m) for 3≤b<q+43.

Construction of Polar Codes Based on Memetic Algorithm

As an emerging coding technique chosen by the 5th generation wireless systems (5G) standardization process, polar coding has attracted the attention of both academia and industry in recent years. The currently used construction methods of polar code are generally based on the reliability of each polarized sub-channel, which directly correspond to the Successive Cancellation (SC) decoding algorithm. However, much less is known about the optimal construction of polar codes under more sophisticated decoding methods such as the Successive Cancellation List (SCL) decoding. There is currently no perfect theory to derive the optimal construction for the SCL decoding to the best of our knowledge, because finding such optimal construction of SCL is considered to be a hard problem for normal choices of the list size. To address this problem, in this paper we turn to evolutionary computation, and propose a memetic algorithm by incorporating prior knowledge obtained from both the channel estimation of the SC-profile and the codeword weight evaluation of the RM-profile, shorted as MA-SCRM. The potential constructions of polar codes are expressed as binary sequences which are processed as the individuals of the population in MA-SCRM. Based on the prior knowledge obtained from the SC-profile and the Reed-Muller-profile, we design special initialization and evolutional operators for polar codes construction. An elitism archive is also introduced to maintain promising constructions and accelerate the convergence of the algorithm. For the SCL decoding, MA-SCRM obtains polar-coding constructions that are more excellent than the SC decoder construction in the sense of block error probability. Compared with the SC decoding construction method, MA-SCRM can reduce the block error probability by up to 48.78% for the original block error rate around <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10^{-2}$</tex-math></inline-formula> , and more improvement can be observed for lower original block error rate.

A Note on Small Weight Codewords of Projective Geometric Codes and on the Smallest Sets of Even Type

A Note on Small Weight Codewords of Projective Geometric Codes and on the Smallest Sets of Even Type

Hadamard matrices of order 36 formed by codewords in some ternary self-dual codes

In this note, we study the existence of Hadamard matrices of order 36 formed by codewords of weight 36 in some ternary near-extremal self-dual codes of length 36.

Classification of Δ-Divisible Linear Codes Spanned by Codewords of Weight Δ

We classify all <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary Δ-divisible linear codes which are spanned by codewords of weight Δ. The basic building blocks are the simplex codes, and for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> = 2 additionally the first order Reed-Muller codes and the parity check codes. This generalizes a result of Pless and Sloane, where the binary self-orthogonal codes spanned by codewords of weight 4 have been classified, which is the case <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> = 2 and Δ = 4 of our classification. As an application, we give an alternative proof of a theorem of Liu on binary Δ-divisible codes of length 4Δ in the projective case.

Design of Polar Codes and PAC Codes for SCL Decoding

The performance of a code under the maximum-likelihood (ML) decoder highly depends on the weight enumerating function (WEF). However, how to compute efficiently the WEF of a polar code or a polarization-adjusted convolutional (PAC) code is still an open problem. For the design of stand-alone polar codes, we consider enumerating the number of minimum-weight non-zero codewords of the polar code. The block error rate (BLER) under the ML decoder can be approximated as a function of the minimum weight of non-zero codewords and its multiplicity. On the other hand, for the design of PAC codes, we consider enumerating the WEF averaged over the ensemble of random PAC codes. The ML-BLER upper bound can be represented as a function of the WEF. The bit-channel selection algorithms for polar codes and PAC codes are proposed, which take both utilization of the polarization effect and the ML decoding performance as selection criteria. Simulation results show that the proposed stand-alone polar codes are competitive when the block length gets larger. Also, simulation results show that the proposed PAC codes yield excellent performance for a wide range of code rates and block lengths and outperform the 5G polar codes.

An Adaptive Chase-Pyndiah Algorithm for Turbo Product Codes

The Chase-Pyndiah algorithm is a soft decoding algorithm for turbo product codes (TPCs) with promising error correction performance. In this letter, we propose an adaptive Chase-Pyndiah algorithm to improve the error correction performance for TPCs with negligible complexity increase. The proposed algorithm adaptively adjusts the weighting factor and reliability factor according to the analog weight of the decision codeword for a component code. Simulation results show that the proposed algorithm can achieve a signal-to-noise ratio (SNR) gain of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.16\sim 0.18$ </tex-math></inline-formula> dB at the bit-error-rate (BER) of 10−6 over additive white Gaussian noise (AWGN) channels.

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

In this letter, we propose to add dynamic frozen bits to underlying polar codes with a Reed-Muller information set with the aim of maintaining the same sub-decoding structure in Automorphism Ensemble (AE) and lowering the Maximum Likelihood (ML) bound by reducing the number of minimum weight codewords. We provide the dynamic freezing constraint matrix that remains identical after applying a permutation linear transformation. This feature also permits to drastically reduce the memory requirements of an AE decoder with polar-like codes having dynamic frozen bits. We show that, under AE decoding, the proposed dynamic freezing constraints lead to a gain of up to 0.25dB compared to the ML bound of the R(3,7) Reed-Muller code, at the cost of small increase in memory requirements.

On the Minimum Weight Codewords of PAC Codes: The Impact of Pre-Transformation

On the Minimum Weight Codewords of PAC Codes: The Impact of Pre-Transformation

On Hard and Soft Decision Decoding of BCH Codes

The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related in recent publications to polar codes. We study the decoding performance of these different BCH codes using information set decoding based on minimal weight codewords of the dual code. We show that information set decoding is possible even in case of a channel without reliability information since the decoding algorithm inherently calculates reliability information. Different BCH codes of the same rate are compared and different decoding performances and complexity are observed. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can possibly be extended to include reliability information of a Gaussian channel for soft decision decoding. We show simulation results for soft decision list information set decoding and compare the performance to other methods.

Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order p^2

The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order p^2, where p is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and antipodal planes. In the Desarguesian case, we can exclude such code words by showing a more general result that no antipodal plane of order at least 3 can be embedded in a Desarguesian projective plane. Furthermore, we use combinatorial arguments to rule out the existence of code words in the dual code of points and lines of an arbitrary projective plane of order p^2, p prime, of weight at most 2p^2-2p+4 using more than two symbols. In particular, this leads to the result that the dual code of the Desarguesian projective plane {{,textrm{PG},}}(2,p^2), pge 5, has minimum weight at least 2p^2-2p+5.