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  
Combinatorial t-designs have been an interesting topic in combinatorics for decades. It is a basic fact that the codewords of a fixed weight in a code may hold a t-design. Till now only a small amount of work on constructing t-designs from codes has been done. 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 satisfying the generalized Kasami
more » ... se, respectively. 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 particular, the codes derived from the dual of binary BCH codes hold five 3-designs when m = 4. 2020 Mathematics Subject Classification: Primary: 05B05, 94B05, 11T23, 11T71.
doi:10.3934/amc.2020106 fatcat:g2kh4id4l5hzvkn5i4yfcfluo4