A search-to-decision reduction for the linear code equivalence problem

We present a polynomial-time reduction from the search variant of the linear code equivalence problem (i.e. the search for a linear isometry between the inputs) to its decisional variant. More precisely, given two linearly equivalent codes $\mathcal C_1,\mathcal C_2 \subseteq \mathbb{F}_q^n$, we sho