An Assmus-Mattson-type approach for identifying 3-designs from linear codes over \(\mathbb{Z}_4\)
From the supports of codewords of a code, which have a same support size \(\tau\), it is sometimes possible to construct a \(t\)-\((n,\tau,\lambda)\)-design. The Assmus-Mattson theorem of \textit{E. F. Assmus, Jr.} and \textit{H. F. Mattson, Jr.} [J. Comb. Theory 6, 122--151 (1969; Zbl 0179.02901)] gives a sufficient condition for the codewords of minimum Hamming weight in a code over a finite field to form a simple \(t\)-design. The authors determine in this article the complete weight enumerator of the Delsarte-Goethals code over \(\mathbb{Z}_4\). They also present an Assmus-Mattson type approach to identify \(t\)-designs in linear codes over \(\mathbb{Z}_4\). This approach, in combination with the complete weight enumerator, is used to show that the codewords of constant Hamming weight in the Goethals code over \(\mathbb{Z}_4\) as well as in the Delsarte-Goethals code over \(\mathbb{Z}_4\) yield 3-designs, possibly with repeated blocks. More precisely, for the Goethals code of length \(2^m\), with odd \(m \geq 5\), the supports of codewords of the same Hamming weight greater than or equal to 8 form a 3-design, possibly with repeated blocks. Similarly, for the Goethals-Delsarte code of length \(2^m\), with odd \(m\geq 5\), the supports of codewords of the same Hamming weight smaller than or equal to \(2^m-3\) form a 3-design, possibly with repeated blocks. A special case of their Assmus-Mattson approach, used for the Hamming weight enumerator, enables them to derive an Assmus-Mattson type theorem for linear codes over \(\mathbb{Z}_4\).
scientific article; zbMATH DE number 2073423