Algorithms for Sampling Self-Orthogonal and Totally Self-Orthogonal Codes in Odd Characteristic
This study addresses the lack of efficient uniform sampling methods for self-orthogonal and totally self-orthogonal linear codes of arbitrary dimension over finite fields by reconstructing sampling algorithms based on mass formulas. It achieves, for the first time, the uniform generation of self-dual codes at arbitrary rates. The concept of "total orthogonality" is introduced to handle equivalence under Galois automorphisms, alongside a proposed sampling scheme for linear codes with prescribed total hull dimension. Furthermore, it is proven that all linear codes possess a constant Hermitian type. This work successfully instantiates code-based cryptographic schemes whose security relies on random self-dual codes, and elucidates the trade-off between extension degree and rate reduction, as well as its necessity in the permuted code equivalence (PCE) problem.