Combinatorial t-designs have been an interesting topic in combinatorics for decades. It was recently reported that the image sets of a fixed size of certain special polynomials may constitute a t-design. Till now only a small amount of work on constructing t-designs from special polynomials has been done, and it is in general hard to determine their parameters. In this paper, we investigate this idea further by using quadratic functions over finite fields, thereby obtain infinite families of 2-designs, and explicitly determine their parameters. The obtained designs cover some earlier 2-designs as special cases. Furthermore, we confirmed Conjecture 3 in Tang (arXiv: 1903.07375, 2019).
BCH codes are an important class of cyclic codes, and have wide applications in communication and storage systems. In this paper, we study the negacyclic BCH code and cyclic BCH code of lengthThe dimensions of negacyclic BCH codes C (n,−1,δ,0) with few nonzeros and C (n,−1,δ,b) with b = 1 are settled. For cyclic BCH code, we give the weight distributions of extended codes C (n,1,δ,1) for δ = δ 1 , δ 2 and the parameters of dual code C ⊥ (n,1,δ,1) for δ 2 ≤ δ ≤ δ 1 .
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