Codes over \(\mathbb {F}_4\) and \(\mathbb {F}_2 \times \mathbb {F}_2\) and theta series of the corresponding lattices in quadratic fields

Using codes defined over \(\mathbb {F}_4\) ${\mathbb{F}}_4$ and \(\mathbb {F}_2 \times \mathbb {F}_2\) ${\mathbb{F}}_2\times {\mathbb{F}}_2$ , we simultaneously define the theta series of corresponding lattices for both real and imaginary quadratic fields \(\mathbb {Q}(\sqrt{d})\) $\mathbb{Q}(\sqrt{d})$ with \(d \equiv 1\mod 4\) $d\equiv 1mod4$ a square-free integer. For such a code, we use its weight enumerator to prove which term in the code’s corresponding theta series is the first to depend on the choice of d. For a given choice of real or imaginary quadratic field, we find conditions on the length of the code relative to the choice of quadratic field. When these conditions are satisfied, the generated theta series is unique to the code’s symmetric weight enumerator. We show that whilst these conditions ensure all non-equivalent codes will produce distinct theta series, for other codes that do not satisfy this condition, the length of the code and choice of quadratic field is not always enough to determine if the corresponding theta series will be unique.