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Infinite families of 2-designs from a class of linear codes related to Dembowski-Ostrom functions [article]

Rong Wang, Xiaoni Du, Cuiling Fan, Zhihua Niu
2019 arXiv   pre-print
Finally, we obtain infinite families of 2-designs from the supports of all codewords with a fixed weight in these codes. Furthermore, the parameters of 2-designs are calculated explicitly.  ...  As we all known, t-designs can be used to derive linear codes over any finite field, as well as the supports of all codewords with a fixed weight in a code also may hold a t-design.  ...  ACKNOWLEDGMENTS The authors would like to thank the anonymous referees for their helpful comments and suggestions, which have greatly improved the presentation and quality of this paper.  ... 
arXiv:1912.07531v1 fatcat:hac5upqowvggrkj37fcnyvbxiu

Infinite families of 2-designs from a class of non-binary Kasami cyclic codes

Rong Wang, ,College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China, Xiaoni Du, Cuiling Fan, ,Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China, ,School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610000, China
2019 Advances in Mathematics of Communications  
In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of 2-designs from the supports of all codewords  ...  It is well known that a linear code over any finite field can be derived from the incidence matrix of a t-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold  ...  Acknowledgements The authors would like to thank the anonymous referees for their helpful comments and suggestions, which have greatly improved the presentation and quality of this paper.  ... 
doi:10.3934/amc.2020088 fatcat:uqgxecvpvzaudfquh3fesy6oa4

Infinite families of 2-designs from two classes of linear codes [article]

Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang
2019 arXiv   pre-print
In this paper, by determining the weight distributions of two classes of linear codes, we derive infinite families of 2-designs from the supports of codewords with a fixed weight in these codes, and explicitly  ...  It is well known that the supports of all codewords with a fixed weight in a code may hold a t-design.  ...  We begin this subsection by the introduction of affine-invariant codes since the two classes of linear codes we investigate are both affine-invariant and will be proved to hold 2-designs.  ... 
arXiv:1903.07459v1 fatcat:wst6kefl4rgbvbp7epzevylhyi

Infinite families of 2-designs from a class of non-binary Kasami cyclic codes [article]

Rong Wang, Xiaoni Du, Cuiling Fan
2019 arXiv   pre-print
In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of 2-designs from the supports of all codewords  ...  with a fixed weight in these codes, and calculate their parameters explicitly.  ...  ACKNOWLEDGMENTS The authors would like to thank the anonymous referees for their helpful comments and suggestions, which have greatly improved the presentation and quality of this paper.  ... 
arXiv:1912.04745v1 fatcat:f5iqjmldevf2rleykdepaxtr44

Affine Invariant Extended Cyclic Codes Over Galois Rings

B.K. Dey, B.S. Rajan
2004 IEEE Transactions on Information Theory  
New classes of afflne invariant BCH and GRM codes over such rings are found.  ...  Afflne invariant extended cyclic codes of length p" over any subring of GR(p",m) for e = 2 with arbitrary p and for p = 2 with arbitrary e are characterized using transform technique.  ...  In this paper, we extend their approach to codes over Galois rings with more general parameters and find new classes of &ne invariant BCH and GRM codes.  ... 
doi:10.1109/tit.2004.825044 fatcat:2y57gwsvn5h5xicz3s5bmmfeaa

Affine invariant extended cyclic codes over Galois rings

Bikash Kumar Dey, B. Sundar Rajan
2003 IEEE International Symposium on Information Theory, 2003. Proceedings.  
New classes of afflne invariant BCH and GRM codes over such rings are found.  ...  Afflne invariant extended cyclic codes of length p" over any subring of GR(p",m) for e = 2 with arbitrary p and for p = 2 with arbitrary e are characterized using transform technique.  ...  In this paper, we extend their approach to codes over Galois rings with more general parameters and find new classes of &ne invariant BCH and GRM codes.  ... 
doi:10.1109/isit.2003.1228174 fatcat:qspm5puqqbbolojjob66vw5qxi

Infinite families of 2-designs from a class of cyclic codes with two non-zeros [article]

Xiaoni Du, Rong Wang, Cuiling Fan
2019 arXiv   pre-print
We then obtain infinite families of 2-designs and explicitly compute their parameters from the supports of all the codewords with a fixed weight in the codes.  ...  It is well known that the supports of all codewords with a fixed weight in a code may give a t-design.  ...  Theorem 1 For each i with A i = 0 in an affine-invariant code C, the supports of the codewords of weight i form a 2-design.  ... 
arXiv:1904.04242v1 fatcat:v7iep2wigvh4tjbgt46udlcvay

Invariance in Property Testing [chapter]

Madhu Sudan
2010 Lecture Notes in Computer Science  
are themselves vector spaces, while their domains are also vector spaces and the properties are invariant under affine transformations of the domain.  ...  We refer to this set of relabellings as the "invariance class" of the property and advocate explicit identification of the invariance class of locally testable properties.  ...  Acknowledgments Thanks to Tali Kaufman, Elena Grigorescu, and Ben-Sasson for their collaboration and comments.  ... 
doi:10.1007/978-3-642-16367-8_12 fatcat:5s4qlkjv3bajnorcwqexgilpja

Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros

Xiaoni Du, ,College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China, Rong Wang, Chunming Tang, Qi Wang, ,Guangxi Key Laboratory of Cryptography and Information Security Guilin University of Electronic Technology Guilin, Guangxi 541004, China, ,School of Mathematics and Information, China West Normal University, Nanchong, Sichuan 637002, China, ,Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
2019 Advances in Mathematics of Communications  
We then obtain infinite families of 2-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters.  ...  In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the tripleerror correcting binary BCH codes, and the other related to the cyclic codes with parameters  ...  Acknowledgments The authors would like to thank the anonymous referees for their helpful comments and suggestions, which have greatly improved the presentation and quality of this paper.  ... 
doi:10.3934/amc.2020106 fatcat:g2kh4id4l5hzvkn5i4yfcfluo4

Linear Codes Of 2-Designs As Subcodes Of The Extended Generalized Reed-Muller Codes [article]

Zhiwen He, Jiejing Wen
2020 arXiv   pre-print
We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes.  ...  The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs.  ...  Acknowledgement The work of the first author was supported by National Natural Science Foundation of China under Grant No. 11771392.  ... 
arXiv:2007.14106v1 fatcat:3fzylwhhvvagjk7bkjju4ivo2u

Infinite families of 2-designs from two classes of binary cyclic codes with three nonzeros [article]

Xiaoni Du, Rong Wang, Chunming Tang, Qi Wang
2019 arXiv   pre-print
We then obtain infinite families of 2-designs from these codes by proving that they are both affine-invariant codes, and explicitly determine their parameters.  ...  In this paper, we determine the weight distributions of two classes of cyclic codes: one related to the triple-error correcting binary BCH codes, and the other related to the cyclic codes with parameters  ...  TWO CLASSES OF CYCLIC CODES AND THEIR t-DESIGNS In this section, we introduce the main results on the weight distributions of two classes of cyclic codes and the corresponding 2-designs.  ... 
arXiv:1903.08153v1 fatcat:xtswbynwm5c6jj6mkvumlk3zt4

On design-theoretic aspects of Boolean and vectorial bent functions [article]

Alexandr Polujan, Alexander Pott
2020 arXiv   pre-print
This implies, that isomorphism of translation designs for Boolean bent functions is a coarser equivalence relation than extended-affine equivalence.  ...  There are two construction methods of designs from Boolean and vectorial bent functions, known as translation and addition designs.  ...  Invariants of equivalence classes and their translation designs are listed in Table B .1(b) and with the structure of the finite field (F 2 n/2 , +, ·) and identify F n 2 with F 2 n/2 ×F 2 n/2 .  ... 
arXiv:2003.12308v2 fatcat:2rpnfzboefcjjbhbp5vnsjoaky

The permutation group of affine-invariant extended cyclic codes

T.P. Berger, P. Charpin
1996 IEEE Transactions on Information Theory  
The extended cyclic codes of length p" , p a prime, which are invariant under the affine-group acting on ffpm, are called affine-invariant codes.  ...  Afterwards we give several tools in order to determine effectively the group of a given code or of some infinite class of codes.  ...  Now we are able to define cyclic codes, extended cyclic codes, and affine-invariant codes by means of a property of their automorphism group: Dejinition 6: A code C of length pm -1 over IF,, where q divides  ... 
doi:10.1109/18.556607 fatcat:ap2c3vku75ftdhizdhw7gje4wi

Design of a novel convex hull based feature set for recognition of isolated handwritten Roman numerals [article]

Nibaran Das, Sandip Pramanik, Subhadip Basu, Punam Kumar Saha, Ram Sarkar, Mahantapas Kundu
2015 arXiv   pre-print
Experiments of convex hull based features for handwritten character recognition are few in numbers. Convex hull of a pattern and the centroid of the convex hull both are affine invariant attributes.  ...  The performance of the designed feature set is tested on the standard MNIST data set, consisting of 60000 training and 10000 test images of handwritten Roman using an MLP based classifier a maximum success  ...  Acknowledgements: Authors are thankful to the "Center for Microprocessor Application for Training Education and Research", "Project on Storage Retrieval and Understanding of Video for Multimedia" of Computer  ... 
arXiv:1501.05494v1 fatcat:jeufd5zmszgxtcmiovnzozavsm

Page 3515 of Mathematical Reviews Vol. , Issue 2004d [page]

2004 Mathematical Reviews  
(D-DORT; Dortmund) Defining sets of cyclic codes invariant under the affine group. (English summary) International Workshop on Coding and Cryptography ( Paris, 2001), 9 pp. (electronic), Electron.  ...  Summary: “We describe the defining sets of extended cyclic codes of length p” over a field and the ring of integers modulo p* admitting the affine group AGL,,(p‘), 1 = mt, as a permutation group.” 2004d  ... 
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