The Permutation Equivalence Problem (PEP) for linear complementary dual (LCD) codes reduces to Graph Isomorphism (GI) through the orthogonal projector onto the code, but this requires a trivial hull. We ask whether the reduction extends when the standard inner product is replaced by a nondegenerate symmetric bilinear form \(M\). We make precise what it means for \(M\) to yield a valid reduction to GI and prove that, for length \(n \geq 3\) and every field size, this holds if and only if \(M\) is permutation-invariant. The reduction then reaches, beyond LCD codes, a class of codes with one-dimensional hull: exactly those whose hull is spanned by a vector with nonzero coordinate sum. We count these codes exactly; for odd \(q\), asymptotically the method covers a fraction \(\prod_{i \geq 2}(1+q^{-i})^{-1}\) of all codes, \(1 + q^{-1}\) times the fraction of LCD codes. A polynomial-time reduction algorithm is given.

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