Cyclic Codes Research Articles

Bounding the Sum of µ-Invariants On Pair Symbol Weights over Some Irreducible Codes

Let Fq be a finite field with q=3 mod 4. Let w be a primitive element of Fq4⁎ and let c(w) be the irreducible cyclic code of length (q4 - 1)/(q - 1) of dimension 4 defined by w. The symbol-pair weight enumerator of C(w) is determined exactly by the invariant µ(w) introduced in Zhu et al. (2022). The determination of good upper and lower bounds on this invariant remains an open problem. Let W be the set of all primitive elements of Fq4⁎. In this paper, by using algebraic and arithmetical methods, particularly those from the theory of algebraic curves over finite fields, we derive effective upper and lower bounds on ΣwεWμ(w).

Explicit Cyclic and Quasi-Cyclic Codes with Optimal, Best Known Parameters and Large Relative Minimum Distances

Explicit Cyclic and Quasi-Cyclic Codes with Optimal, Best Known Parameters and Large Relative Minimum Distances

New ternary LCD cyclic and BCH codes

New ternary LCD cyclic and BCH codes

Constacyclic codes over the non-chain finite commutative ring 4 2 2 [ , ] , , u v 〈u − u v uv〉

This paper considers (1+2u) -constacyclic and (1+2u) -skew constacyclic codes over the ring R = Z4[u,v]/<u2 – u, v2, uv>, and some Gray maps over this ring are presented. The Gray images of these codes are shown to be cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes. In addition, (1+2u) constacyclic codes are given with images that are new Z4 codes according to the online database of linear Z4 codes [3].

Two classes of constacyclic codes with a square-root-like lower bound

Two classes of constacyclic codes with a square-root-like lower bound

Prime-Factor GFFT Architecture for Fast Frequency Domain Decoding of Cyclic Codes

Prime-Factor GFFT Architecture for Fast Frequency Domain Decoding of Cyclic Codes

On the squares of LCD cyclic codes and their complements: study of several families and analyzing their parameters

On the squares of LCD cyclic codes and their complements: study of several families and analyzing their parameters

On double cyclic codes over $ \mathbb{Z}_2+u\mathbb{Z}_2 $

<abstract><p>In this paper, we introduced double cyclic codes over $ R^r\times R^s $, where $ R = \mathbb{Z}_{2}+u\mathbb{Z}_{2} = \{0, 1, u, 1+u\} $ is the ring with four elements and $ u^2 = 0 $. We first determined the generator polynomials of $ R $-double cyclic codes for odd integers $ r $ and $ s $, then gave the generators of duals of free double cyclic codes over $ R^r\times R^s $. By defining a linear Gray map, we looked at the binary images of $ R $-double cyclic codes and gave several examples of optimal parameter binary linear codes obtained from $ R $-double cyclic codes. Moreover, we studied self-dual $ R $-double cyclic codes and presented an example of a self-dual $ R $-double cyclic code.</p></abstract>

Algebraic structure of cyclic codes over M3(Fp)

Let Fp be the finite field of order p and M3(Fp) the ring of 3 × 3 matrices over Fp, where p is a prime. For certain prime p, we determine the complete algebraic properties of cyclic codes of length N (p | N) over M3(Fp). We define an isometry from M3(Fp) to Fp3 + eFp3 + e2Fp3, where e3 = 1. As an outcome, we derive numerous optimal and good linear F8 codes induced from F8 -images of cyclic codes over M3(F2).

A Class of Ternary Optimal Cyclic Codes with Minimum Distance 4

A Class of Ternary Optimal Cyclic Codes with Minimum Distance 4

Constructions of Binary Constant-Weight Quasi-Cyclic Codes and Quasicyclically Permutable Codes

A theorem is proven showing how to obtain a constant-weight binary quasi-cyclic code from a pr-ary linear cyclic code, where p is a prime and r is a positive integer, r > 1, by using a representation of the elements of a Galois field, GF(pr), as cyclic shifts of a binary pr-tuple. From this theorem, constructions are derived for two classes of constant-weight binary quasi-cyclic codes. These two classes are shown to achieve the Johnson upper bound on the number of codewords asymptotically for long block lengths. A quasicyclically permutable (QCP) code is a binary code such that the codewords are quasicyclically distinct and have cyclic order equal to the code block length. A technique is described for selecting virtually the maximum number of cyclically distinct codewords of full cyclic order from Reed-Solomon (RS) codes and from Berlekamp-Justesen (BJ) codes, both known to be maximum distance separable codes. Those cyclically distinct codewords of full cyclic order from RS codes and from BJ codes are mapped to binary to produce two classes of asymptotically optimum constant-weight quasicyclic codes and two classes of asymptotically optimum constant weight QCP codes. An application of QCP codes is introduced to construct protocol-sequence sets for the M-active-out-of-T users collision channel without feedback, allowing more users than strict cyclically permutable codes with the same block length and minimum distance.

Parameters of some BCH codes over $ \mathbb{F}_q $ of length $ \frac{q^m-1}{2} $

BCH codes, known for their error-correcting capability and efficient encoding and decoding algorithms, are an important class of cyclic codes over finite fields. Currently, BCH codes have been widely applied in consumer devices, communication systems, and data storage systems. However, there is still a lot of unsolved work in exploring the parameters of BCH codes, such as the dimensions and minimum distances. The objective of this paper is to study narrow-sense BCH codes of length $ \frac{q^m-1}{2} $ over $ \mathbb{F}_q $. The parameters of narrow-sense BCH codes of such length with designed distances $ \delta_i = \frac{q^m-q^{m-1}-q^{\lfloor\frac{m-3}2\rfloor+i}-1}{2} $ and $ \delta_{\lfloor\frac{m+6}{4}\rfloor+t} = \frac{q^m-q^{m-1}-q^\frac{3m-1}{4}-q^\frac{m+1}{2}-q^{\frac{m-11}{4}+t}-1}{2} $ are presented, respectively, where $ 1\le i\le \lfloor\frac{m+6}{4}\rfloor $ and $ 2\le t\le 4 $.

Retraction: Xianhong Xie, Sen Wang, Yi Ouyang and Lanqiang Li. A note on subfield subcodes and trace codes of cyclic codes. Advances in Mathematics of Communications, 2024, Early Access article, doi: 10.3934/amc.2024023

Retraction: Xianhong Xie, Sen Wang, Yi Ouyang and Lanqiang Li. A note on subfield subcodes and trace codes of cyclic codes. Advances in Mathematics of Communications, 2024, Early Access article, doi: 10.3934/amc.2024023

Infinite families of 2-designs from a class of affine-invariant codes

<p style='text-indent:20px;'>In this paper, we first introduce a class of cyclic codes and their related linear codes which are affine-invariant. Then we obtain infinite families of 2-designs from the affine-invariant codes. And the parameters of the 2-designs are determined by considering the weight distribution of the linear codes.</p>

New Quantum Codes from the Images of Cyclic Codes

New Quantum Codes from the Images of Cyclic Codes

On the ℓ-DLIPs of codes over finite commutative rings

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of ℓ-dimension linear intersection pairs (ℓ-DLIPs) of codes over a finite commutative ring (R), for some positive integer ℓ. In this paper, we study ℓ-DLIP of codes over R in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) ℓ-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring which helps us to get an important characterization of ℓ-DLIP of constacyclic codes. Finally, the ℓ-DLIP of constacyclic codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.

Additive conjucyclic codes over a class of Galois rings

As a tool towards quantum error correction, additive conjucyclic codes have gained great attention. But, their algebraic structure is completely unknown over finite fields (except Fq2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {F}}_{q^2}$$\\end{document}) as well as rings. In this article, we investigate the structure of additive conjucyclic codes over Galois rings GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document}, where r≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r\\ge 2$$\\end{document} is an integer. We develop a one-to-one correspondence between the family of additive conjucyclic codes of length n over GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document} and the family of linear cyclic codes of length 2n over Z2r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2^r}$$\\end{document}. This correspondence helps to obtain additive conjucyclic codes over GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document} via known linear cyclic codes over Z2r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2^r}$$\\end{document}. We prove that the trace dual CTr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {C}}^{Tr}$$\\end{document} of an additive conjucyclic code C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathscr {C}}$$\\end{document} is also an additive conjucyclic code. Moreover, we derive a necessary and sufficient condition of additive conjucyclic codes to be self-dual. We further propose a technique for constructing linear cyclic codes over Z2r\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}_{2^r}$$\\end{document} contained in additive conjucyclic codes over GR(2r,2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$GR(2^r,2)$$\\end{document}. Last but not least, we explicitly derive the generator matrices for these codes.

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.

Perceived constraints to participating in wildlife-related recreation

Wildlife-related recreationists play an important role in conservation. Understanding constraints to wildlife-related activities is critical for maintaining or increasing participation in activities like birdwatching and hunting. A mail-out survey was administered to a generalized sample representative of U.S. residents (i.e., not specific to birdwatching or hunting) in early 2017 to determine what would limit them from participating in birdwatching and hunting (n = 1030). We employed a concurrent nested mixed-methods design: open-ended responses were thematically coded qualitatively in two distinct cycles (i.e., inductive, and then mixed inductive-deductive coding), and then the probability of expressing the second cycle codes was quantitatively modeled using multinomial logit models for the respective activities. Doing so empirically determined various groups’ constraints that are important to recruitment, retention, and reactivation (R3) efforts for birdwatching and hunting. We found that the likelihood of experiencing unique constraints varied based on sociodemographic characteristics, and these relationships differed between birdwatching and hunting. Gender had a limited effect on constraints to birdwatching but was a strong indicator of intrapersonal constraints and limitations to involvement in hunting. The likelihood of expressing structural constraints decreased with age for both activities. Possessing strong social ties to the activities tended to reduce the likelihood of expressing constraints overall but this was especially true for hunting. Our findings inform R3 efforts for wildlife-related recreation and provide direct results that organizations can apply in seeking to help Americans negotiate constraints and increase and diversify participation in wildlife-related recreation and conservation behavior. Management implicationsOur results have implications for recruitment, retention, and reactivation (R3) efforts to increase participation in birdwatching and hunting. Those seeking to increase participation in birdwatching should prioritize younger segments of the population because they expressed more constraints than older segments except for limited capacity to participate. For hunting, gender was a key variable in determining expressed constraints; identifying as female increased the likelihood of expressing intrapersonal constraints and limitations in involvement. For both activities, possessing strong social ties (i.e., family, and close friends) to both activities were effective in decreasing the likelihood of expressing many constraints. Organizations seeking to recruit new participants to birdwatching or hunting should develop programs that encourage existing participants to involve friends and family with the activities and foster interest in the activities among non-participants.

Several classes of optimal cyclic codes with three zeros

Several classes of optimal cyclic codes with three zeros