Infinite families of linear codes supporting more t-designs
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Tang and Ding [IEEE IT 67 (2021) 244-254] studied the class of narrow-sense BCH codes 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 infinite families of 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 the codewords of weight 7 in 𝒞_(q,q+1,4,1) support 4-designs when m ⩾ 5 is odd and 3-designs when m ⩾ 4 is even, which provide infinite classes of simple t-designs with new parameters. Another significant class of t-designs we produce in this paper has supplementary 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,λ) 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 automorphism group constructed by Alltop in 1969.
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