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The weight distribution of some irreducible cyclic codes

Anuradha Sharma, Gurmeet K. Bakshi
2012 Finite Fields and Their Applications  
In this paper, we explicitly determine the weight distribution of all the irreducible cyclic codes of length p m over F q from their generating polynomials in three different cases, when (i) the multiplicative  ...  order of q modulo p m is φ(p m ), (ii) the multiplicative order of q modulo p m is a power of p, and (iii) the multiplicative order of q modulo p m is twice a power of p.  ...  Acknowledgments The authors are grateful to the anonymous referees for their comments and suggestions which helped to write the paper in the present form.  ... 
doi:10.1016/j.ffa.2011.07.002 fatcat:yuc7cli5uzarnfgvxeozcqrefq

The weight distribution of families of reducible cyclic codes through the weight distribution of some irreducible cyclic codes

Gerardo Vega, ,Dirección General de Cómputo y de Tecnologías de Información y Comunicación, Universidad Nacional Autónoma de México, 04510 Ciudad de México, Mexico, Jesús E. Cuén-Ramos, ,Posgrado en Ciencias Matemáticas, Universidad Nacional Autónoma de México, 20059 Ciudad de México, Mexico
2019 Advances in Mathematics of Communications  
The calculation of the weight distribution for some reducible cyclic codes can be reduced down to the corresponding one of a particular kind of irreducible cyclic codes.  ...  In fact, as will be shown here, the weight distribution of some families of reducible cyclic codes, recently reported in several works ([2, 5, 7, 11, 12]), and that of others not previously reported, can  ...  Acknowledgments The authors would like to express their gratitude to the anonymous referees for their valuable suggestions.  ... 
doi:10.3934/amc.2020059 fatcat:jgn4gwuowfb3rktj6c72cpuw7m

Determining the full weight distribution of any irreducible cyclic code over any finite field of dimension two [article]

Gerardo Vega
2016 arXiv   pre-print
In fact, the relevance of our results is that by means of them we can actually directly determine the full weight distribution of any irreducible cyclic code, of dimension one or two, just by knowing its  ...  In this note, we are going to use a characterization of all semiprimitive two-weight irreducible cyclic codes over any finite field, recently presented in [8], in order to determine the full weight distribution  ...  Since the weight distributions of these two types of irreducible cyclic codes are already known, we can then determine the full weight distribution of any irreducible cyclic code over any finite field  ... 
arXiv:1606.08510v1 fatcat:afqwioemgrf2ngeyejflch23eq

Primitive Idempotents and Weight Distributions of Irreducible Cyclic Codes of Length 3𝑙𝑚

Sunil Kumar, Manju Pruthi, Rahul
2019 Zenodo  
We give the weight distributions of all irreducible cyclic codes of length 3𝑙𝑚 over 𝔽𝑞 .  ...  Moreover, they obtain the dimensions and the minimum hamming distances of all irreducible cyclic codes we give the weight distribution of all irreducible cyclic codes and their dual codes of length 3  ...  In the following, we give the weight distributions of irreducible cyclic codes over generated by the primitive idempotents in / 3 − 1 .  ... 
doi:10.5281/zenodo.2578099 fatcat:xo4rztmoonanxbqkrsjh4vyvaa

Optimal Sets of Frequency Hopping Sequences From Linear Cyclic Codes

Cunsheng Ding, Yang Yang, Xiaohu Tang
2010 IEEE Transactions on Information Theory  
Optimal sets of frequency hopping sequences are constructed with a class of irreducible cyclic codes. As a byproduct, the weight distribution of a subclass of irreducible cyclic codes is determined.  ...  Index Terms-Cyclic codes, direct sequence spread spectrum, frequency hopping sequence, frequency hopping spread spectrum, irreducible cyclic codes.  ...  ACKNOWLEDGMENT The authors would thank the two anonymous reviewers and the Associate Editor, M. Parker, for their comments that improved the presentation of this paper.  ... 
doi:10.1109/tit.2010.2048504 fatcat:g25yx4r3trhvxktvqspwansvxi

The minimum distance of the duals of binary irreducible cyclic codes

Cunsheng Ding, T. Helleseth, H. Niederreiter, Chaoping Xing
2002 IEEE Transactions on Information Theory  
Irreducible cyclic codes have been an interesting subject of study for many years. The weight distribution of some of them have been determined.  ...  In this paper, we determine the minimum distance and certain weights of the duals of binary irreducible cyclic codes.  ...  ACKNOWLEDGMENT The authors thank the referee and Xuesong Wang for their helpful comments and suggestions that improved this paper.  ... 
doi:10.1109/tit.2002.802593 fatcat:hxhvk53pnngsxd5d6shv7o5sfq

The b-symbol weight distributions of all semiprimitive irreducible cyclic codes [article]

Gerardo Vega
2022 arXiv   pre-print
The purpose of this paper is to simplify and generalize the results of Zhu et al., and obtain the b-symbol weight distributions of all one-weight and two-weight semiprimitive irreducible cyclic codes.  ...  Up to a new invariant μ(b), the complete b-symbol weight distribution of a particular kind of two-weight irreducible cyclic codes, was recently obtained by Zhu et al. [Des.  ...  Introduction Up to a new invariant µ(b), the complete b-symbol weight distribution of some irreducible cyclic codes was recently obtained in [7] .  ... 
arXiv:2206.03459v1 fatcat:fkfayqsbgnhfnpd562hlt3z2am

Hamming weights in irreducible cyclic codes

Cunsheng Ding, Jing Yang
2013 Discrete Mathematics  
The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible  ...  cyclic codes.  ...  Acknowledgments The authors are grateful to the reviewers for their comments that improved the presentation and quality of this paper.  ... 
doi:10.1016/j.disc.2012.11.009 fatcat:rqcxwxpfvfdzbb567io3ro2fh4

Optimal three-weight cyclic codes whose duals are also optimal [article]

Gerardo Vega, Félix Hernández
2021 arXiv   pre-print
Lastly, as an application of the Krawtchouck polynomials, we obtain the weight distribution of the dual codes.  ...  Here we present a new class of optimal three-weight cyclic codes of length q+1 and dimension 3 over any finite field F_q, and show that the nonzero weights are q-1, q, and q+1.  ...  Hernández is a PhD student at the Posgrado en Ciencia e Ingeniería de la Computación, Universidad Nacional Autónoma de México,  ... 
arXiv:2107.04579v1 fatcat:nvho4k6e2jgjnkngx7mxfe7wam

Hamming Weights in Irreducible Cyclic Codes [article]

Cunsheng Ding, Jing Yang
2011 arXiv   pre-print
The objectives of this paper are to survey and extend earlier results on the weight distributions of irreducible cyclic codes, present a divisibility theorem and develop bounds on the weights in irreducible  ...  Irreducible cyclic codes are an interesting type of codes and have applications in space communications. They have been studied for decades and a lot of progress has been made.  ...  distributions of irreducible cyclic codes. • Extensions and generalizations of earlier results on the weight distributions of irreducible cyclic codes (Theorems 24, 22, 16, 18, 19, 20, and 21). • A complete  ... 
arXiv:1108.3887v1 fatcat:fo5yx47nrbghlaiffponcyjfji

Recent progress on weight distributions of cyclic codes over finite fields

Hai Q. Dinh, Chengju Li, Qin Yue
2015 Journal of Algebra Combinatorics Discrete Structures and Applications  
Furthermore, we discuss the more general case of constacyclic codes and give some equivalences to characterize their weight distributions.  ...  In coding theory it is often desirable to know the weight distribution of a cyclic code to estimate the error correcting capability and error probability.  ...  Acknowledgment: The authors are very grateful to the reviewers and the editor for their valuable comments and suggestions that improved the quality of this paper.  ... 
doi:10.13069/jacodesmath.36866 fatcat:ga4rybodfvc7dahch6vqmwfokm

Several Families of Irreducible Constacyclic and Cyclic Codes [article]

Zhonghua Sun and Xiaoqiang Wang and Cunsheng Ding
2022 arXiv   pre-print
The weight distributions of these irreducible constacyclic codes and the parameters of their duals are settled.  ...  Several families of irreducible constacyclic codes with a few weights and several families of optimal constacyclic codes are constructed.  ...  Acknowledgements The authors are very grateful to Prof. Anuradha Sharma for providing reference [27] . All the code examples in this paper were computed with the Magma software package.  ... 
arXiv:2207.01864v1 fatcat:ocun3imvgbactdrtuqyyl4bxhq

Two-Weight Codes and Second Order Recurrences

Minjia Shi, Zhongyi Zhang, Patrick Solé
2019 Chinese journal of electronics  
We compute their weight distribution, and give a condition on the roots of their check polynomials for them to be maximum distance separable code.  ...  Cyclic codes of dimension 2 over a finite field are shown to have at most two nonzero weights.  ...  A cyclic code is irreducible if its check polynomial h(x) is irreducible over the alphabet field. A code is two-weight if it has only two non-zero weights. In Ref.  ... 
doi:10.1049/cje.2019.07.001 fatcat:mkxu3xtq5bgtjifte4osltgg6i

The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs

Ricardo A. Podestá, Denis E. Videla
2021 Advances in Mathematics of Communications  
We use known characterizations of generalized Paley graphs which are Cartesian decomposable to explicitly compute the spectra of the corresponding associated irreducible cyclic codes.  ...  As applications, we give reduction formulas for the number of rational points in Artin-Schreier curves defined over extension fields and to the computation of Gaussian periods.  ...  . since c = 12. ♦ Remark 4.3.The weight distribution of irreducible cyclic codes constructed from exceptional 2-weight irreducible cyclic codes can be obtained from Theorem 2.2 and from the spectrum of  ... 
doi:10.3934/amc.2021002 fatcat:rhghomblujahdcr5ecnjzpy5dm

The Weight Distribution of a Class of Cyclic Codes Related to Hermitian Forms Graphs

Shuxing Li, Sihuang Hu, Tao Feng, Gennian Ge
2013 IEEE Transactions on Information Theory  
The determination of weight distribution of cyclic codes involves evaluation of Gauss sums and exponential sums.  ...  In this note, we determine the weight distribution of a class of reducible cyclic codes whose dual codes may have arbitrarily many zeros.  ...  ACKNOWLEDGEMENT The authors express their gratitude to the two anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of this paper, and to Dr.  ... 
doi:10.1109/tit.2013.2242957 fatcat:xgoelpxjwzdw7hep4jgpemcooy
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