An infinite family of linear codes supporting 4-designs

01/01/2020

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by   Chunming Tang, et al.

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The first linear code supporting a 4-design was the [11, 6, 5] ternary Golay code discovered in 1949 by Golay. In the past 71 years, sporadic linear codes holding 4-designs or 5-designs were discovered and many infinite families of linear codes supporting 3-designs were constructed. However, the question as to whether there is an infinite family of linear codes holding an infinite family of t-designs for t≥ 4 remains open for 71 years. This paper settles this long-standing problem by presenting an infinite family of BCH codes of length 2^2m+1+1 over GF(2^2m+1) holding an infinite family of 4-(2^2m+1+1, 6, 2^2m-4) designs. Moreover, an infinite family of linear codes holding the spherical design S(3, 5, 4^m+1) is presented.