An Infinite Family of Linear Codes Supporting 4-Designs
Chunming Tang, Cunsheng Ding
Abstract
The question as to whether there exists an infinite family of near MDS codes holding an infinite family of t-designs for t ≥ 2 was answered in the recent paper [Infinite families of near MDS codes holding t-designs, IEEE Trans. Inf. Theory 66(9) (2020)], where an infinite family of near MDS codes holding an infinite family of 3-designs and an infinite family of near MDS codes holding an infinite family of 2-designs were presented, but no infinite family of linear codes holding an infinite family of 4-designs was presented. Hence, the question as to whether there is an infinite family of linear codes holding an infinite family of 4-designs remains open for 71 years. This paper settles this longstanding problem by presenting an infinite family of BCH codes of length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+1</sup> +1 over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m+1</sup> ) holding an infinite family of 4-(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m+1</sup> + 1, 6, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sup> - 4) designs. This paper also provides another solution to the first question, as some of the BCH codes presented in this paper are also near MDS. Moreover, an infinite family of linear codes holding the spherical geometry design S(3, 5, 4 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> + 1) is presented. The new direction of searching for t-designs with elementary symmetric polynomials will be further advanced.