Infinite Families of Linear Codes Supporting More t-Designs
Qianqian Yan, Junling Zhou · IEEE Transactions on Information Theory · 2022
Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes$\mathcal {C}_{(q,q+1,4,1)}$and their dual codes with$q=2^{m}$and established that the codewords of the minimum (or the second minimum) weight in these codes support 4-designs or 3-designs. Motivated by this, we further investigate the codewords of the next adjacent weight in such codes and discover more infinite classes of$t$-designs with$t=3,4$. In particular, we prove that codewords of weight 7 in$\mathcal {C}_{(q,q+1,4,1)}$support 4-designs for odd$m \geqslant 5$and they support 3-designs for even$m \geqslant 4$, which provide infinite classes of simple$t$-designs with new parameters. Another significant class of$t$-designs we produce in this paper has complementary designs with parameters 4-$(2^{2s+1}+ 1,5,5)$; these designs have the smallest index among all the known simple 4-$(q+1,5,\lambda)$designs derived from codes for prime powers$q$; and they are further proved to be isomorphic to the 4-designs admitting the projective general linear group PGL$(2,2^{2s+1})$as the automorphism group constructed by Alltop in 1969.