New 3-designs from Goethals codes over \(Z_4\) (Q1841929)

The support of a vector \(c = (c_1,c_2,\dots{},c_n)\) is the subset of \(\{1,2,\dots{},n\}\) given by \(\{j\mid c_j \neq 0\}\). A vector is denoted to be of the type \(1^{n_1}2^{n_2}3^{n_3}0^{n_0}\) if \(j\) occurs \(n_j\) times, \(j=0,1,2,3\), as a component. The codewords of minimum Lee weight in the Goethals code \({\mathcal G}_m\) for any odd integer \(m\) are one of the types \(1^{4}2^{1}3^{2}0^{n-7}\) or \(1^{6}2^{1}3^{0}0^{n-7}\) or \(2^{4}0^{n-4}\) (or their negatives) which has support size \(7\). In this paper the authors construct a \(3\)-\((2^5,7,105)\) design from the support of the codewords of type \(1^{4}2^{1}3^{2}0^{n-7}\) as well as a \(3\)-\((2^5,7,7)\) design from the support of the codewords of type \(1^{6}2^{1}3^{0}0^{n-7}\) for \(m=5\). They further construct a \(3\)-\((2^7,7,560)\) design from the support (size \(7\)) of the codewords obtained by combining the codewords for \(m=7\) of both the types \(1^{4}2^{1}3^{2}0^{n-7}\) and \(1^{6}2^{1}3^{0}0^{n-7}\). Both the designs and their construction method are new.