Infinite Families of Linear Codes Supporting More t-Designs

Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of BCH codes C-(q,C-q+1,C-4,C-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 C-(q,C- q+1,C-4,C-1) support 4-designs for odd m >= 5 and they support 3-designs for even m >= 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.