3-designs from the Z₄-goethals codes via a new kloosterman sum identity

Abstract

Recently, active research has been performed on constructing t-designs from linear codes over Z₄. In this paper, we will construct a new simple 3 − (2^m, 7, 14/3 (2^m − 8)) design from codewords of Hamming weight 7 in the Z₄-Goethals code for odd m ≥ 5. For 3 arbitrary positions, we will count the number of codewords of Hamming weight 7 whose support includes those 3 positions. This counting can be simplified by using the double-transitivity of the Goethals code and divided into small cases. It turns out interestingly that, in almost all cases, this count is related to the value of a Kloosterman sum. As a result, we can also prove a new Kloosterman sum identity while deriving the 3-design.