An Infinite Family of Linear Codes Supporting 4-Designs

The question as to whether there exists an infinite family of near MDS codes holding t-designs for t ? 2 was answered in the recent paper [Infinite families t-designs, IEEE Trans. Inf. Theory 66(9) (2020)], where 3-designs and 2-designs were presented, but no linear 4-designs presented. Hence, is remains open 71 years. This settles this longstanding problem by presenting BCH length m+1 +1 over GF(2 xmlns:xlink="http://www.w3.org/1999/xlink">2m+1 ) 4-(2 + 1, 6, xmlns:xlink="http://www.w3.org/1999/xlink">2m - 4) designs. also provides another solution first question, some presented are MDS. Moreover, spherical geometry design S(3, 5, 4 xmlns:xlink="http://www.w3.org/1999/xlink">m 1) new direction searching with elementary symmetric polynomials will be further advanced.