The permutation equivalence problem (PEP) for linear codes is fundamental to both coding theory and modern cryptography. The support splitting algorithm (SSA) is the most effective practical approach, with efficiency critically dependent on hull dimensions. SSA requires hulls that remain invariant under coordinate permutations, but can use alternative inner products to potentially reduce hull dimensions. We provide a complete characterization: permutation-invariant hulls correspond precisely to bilinear forms with structure matrices of the form aI + bJ. We prove that these alternative hulls can reduce dimensions by at most one, and establish conditions for when reduction occurs. While this bound limits the impact on cryptographic security, the improvement translates to a factor q speedup in enumeration complexity -- significant for code classification and cryptanalysis refinement. Our results apply to schemes like LESS and SPECK, where accurate complexity assessment is crucial for parameter selection.

We give new characterizations of the largest minimum weights among LCD codes of large dimensions. Using the characterizations, we completely determine the largest minimum weights among binary LCD codes of length n and dimension$n-6$, ternary LCD codes of length n and dimension$n-5$and quaternary Hermitian LCD codes of length n and dimension$n-4$for arbitrary n. We also determine the largest minimum weights among binary LCD codes of length n and dimension$n-7$, ternary LCD codes of length n and dimension$n-6$and quaternary Hermitian LCD codes of length n and dimension$n-5$with only some exceptions n.