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Published in:
14-06-2023
Lattices in real quadratic fields and associated theta series arising from codes over \({\textbf{F}}_4\) and \({\textbf{F}}_2 \times {\textbf{F}}_2\)
Published in: Designs, Codes and Cryptography | Issue 10/2023
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Abstract
Let \(K = {\textbf{Q}}(\sqrt{d})\) with \(d>0\) square-free and let \({\mathcal {O}}_{K}\) denote the ring of integers of K. Let \({\mathcal {C}} \subset {\mathcal {R}}^{n}\) be a linear code where \({\mathcal {R}}\) is \({\textbf{F}}_4\) if \(d \equiv 5 \pmod {8}\) and \({\textbf{F}}_2 \times {\textbf{F}}_2\) if \(d \equiv 1 \pmod {8}\). One has a surjective ring homomorphism \(\rho : {\mathcal {O}}_{K}^{n} \rightarrow {\mathcal {R}}^n\) given by reduction modulo \((2{\mathcal {O}}_{K})^{n}\). The inverse image \(\Lambda ({\mathcal {C}}):=\rho ^{-1}({\mathcal {C}})\) is a lattice associated to the code \({\mathcal {C}}\). One can associate to \(\Lambda _{{\mathcal {C}}}\) a theta series \(\Theta _{\Lambda _{d}({\mathcal {C}})}\). In this paper we consider how the theta series varies as one varies the value d. In particular, we show that for \(d, d'\) with \(d>d'\), one has \( \Theta _{\Lambda _{d}({\mathcal {C}})} = \Theta _{\Lambda _{d^\prime }({\mathcal {C}})} + O\left( q^{\frac{d^\prime +1}{2}} \right) .\)