The Permutation Equivalence Problem (PEP) for linear codes is a fundamental problem in coding theory and cryptography. A recent reduction shows that PEP for Linear Complementary Dual (LCD) codes reduces to Graph Isomorphism (GI) via orthogonal projectors, but is restricted to codes with trivial hull. We prove that this approach extends to bilinear forms $M = aI + bJ$, and that no other nondegenerate symmetric form yields a valid reduction. A code is reducible if and only if its hull dimension is at most $1$ with an explicit condition on the hull vector; in characteristic $2$, only LCD codes are reducible. This establishes the closure of the orthogonal projector method. We derive exact enumeration formulas via character sums over quadratic forms and provide a polynomial-time reduction algorithm.

翻译：线性码的置换等价问题(PEP)是编码理论与密码学中的基本问题。近期一项归约表明，通过正交投影算子可将线性互补对偶(LCD)码的PEP问题归约到图同构(GI)问题，但该归约仅适用于平凡核的码。我们证明该方法可推广至双线性形式$M = aI + bJ$，且不存在其他非退化对称形式能产生有效归约。码可归约当且仅当其核维数至多为$1$且满足核向量的显式条件；在特征为$2$的域上，仅LCD码具有可归约性。这确立了正交投影算子方法的闭包性质。我们通过二次型特征和导出精确计数公式，并给出多项式时间归约算法。