We study the complexity of the Code Equivalence Problem on linear error-correcting codes by relating its variants to isomorphism problems on other discrete structures—graphs, lattices, and matroids. Our main results are a fine-grained reduction from the Graph Isomorphism Problem to the Linear Code Equivalence Problem over any field \(\mathbb {F}\), and a reduction from the Linear Code Equivalence Problem over any field \(\mathbb {F}_p\) of prime, polynomially bounded order p to the Lattice Isomorphism Problem. Both of these reductions are simple and natural. We also give reductions between variants of the Code Equivalence Problem, and study the relationship between isomorphism problems on codes and linear matroids.

我们通过将线性纠错码上的代码等价问题与其他离散结构（图、格子和矩阵）上的同构问题相关联，研究了线性纠错码上代码等价问题的复杂性。我们的主要结果是在任何域 \（\mathbb {F}\） 上从图同构问题精细简化为线性码等价问题，以及从素数多项式有界阶 p 的任何域 \（\mathbb {F}_p\） 上的线性码等价问题简化为晶格同构问题。这两种减少都是简单而自然的。我们还给出了代码等价问题变体之间的约简，并研究了代码和线性矩阵上的同构问题之间的关系。