Codes over \(\mathbb F_4\), Jacobi forms and Hilbert-Siegel modular forms over \(\mathbb Q(\sqrt 5)\)
Let \({\mathbb F}_4=\{0,1,\omega,\omega^2\}\) denote the field of four elements and let \(C\) be a code over \({\mathbb F}_4\). For a vector \(x\in {\mathbb F}_4\) the Lee weight is defined as \(n_1(x)+2(n_{\omega}(x)+n_{\omega^2}(x))\), where \(n_a(x), a\in {\mathbb F}_4\) denotes the number of coordinates of \(x\), which are equal to \(a\). For a code \(C\) the complete weight enumerator is the polynomial \[ cwe_C(y_0,y_1,y_{\omega},y_{\omega^2}) = \sum_{v\in C} \prod_{a\in {\mathbb F}_4} y_a^{n_a(v)}. \] First MacWilliam's identity for a complete weight enumerator is generalized for the present setting. A self-dual code over \({\mathbb F}_4\) is said to be Type II, if the Lee weight of every codeword is a multiple of \(4\) and Type I otherwise. There are given several examples of Type I and Type II codes together with their complete weight enumerators. It is defined a group \(G'\) of \(4\times 4\) complex matrices of order 7680 and proved that the complete weight enumerator of self-dual codes and self-dual codes of Type II are invariant polynomial under certain subgroups of \(G'\). There are constructed integral lattices induced from self-dual codes over \({\mathbb F}_4\) and Jacobi forms over the real quadratic number field \(K={\mathbb Q}(\sqrt{5})\) and Hilbert modular forms over \(K\) from their various weight enumerators.
scientific article; zbMATH DE number 2165406