Algorithms for Sampling Self-Orthogonal and Totally Self-Orthogonal Codes in Odd Characteristic

2026-10-06

AI Summary

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.

Abstract

We give an algorithm that samples uniformly random linear codes of any hull dimension and type over finite fields of odd characteristic, implying the first algorithm for sampling uniformly random self-orthogonal codes of any rate, including self-dual codes. Our algorithm is a re-visitation of the algorithm given by Albrecht, Benčina and Lai (EC'25), using the mass formulae proven by Li, Shi and Ling (IEEE Trans. Inf. Theory 71(1)). This allows us to instantiate code-based cryptographic schemes that rely on the hardness of Permutation Code Equivalence (PCE) on `random' self-dual codes for security and that were previously unable to sample them. Building on the observation by Bardet, Otmani and Saeed-Taha (ISIT'19) that Euclidean orthogonality is insufficient when considering PCE over finite extension fields due to non-trivial Galois automorphisms, we study the behaviour of what we call total orthogonality, that is orthogonality with respect to all induced Galois geometries simultaneously. We characterise total orthogonality of vectors and codes, and give an algorithm that samples linear codes with a total hull of a prescribed dimension; a subcode that acts as the hull in all Galois geometries of the ambient space. The algorithm incurs a rate decrease by a factor equal to the extension degree, however, we argue why this may be necessary in the context of PCE and explore how it limits the practicality of our algorithm. We consider the notion of Galois type of a linear code when Galois hulls are symmetric and show that all linear codes have constant Hermitian type.

Problem

• Self-Orthogonal Codes
• Permutation Code Equivalence
• Odd Characteristic
• Totally Self-Orthogonal Codes
• Code Sampling

Innovation

• Self-Orthogonal Codes
• Permutation Code Equivalence
• Total Orthogonality
• Galois Hull
• Code Sampling Algorithm