The linear code equivalence (LCE) problem is shown to be equivalent to the point set equivalence (PSE) problem, i.e., the problem to check whether two sets of points in a projective space over a finite field differ by a linear change of coordinates. For such a point set $\mathbb{X}$, let $R$ be its homogeneous coordinate ring and $\mathfrak{J}_{\mathbb{X}}$ its canonical ideal. Then the LCE problem is shown to be equivalent to an algebra isomorphism problem for the doubling $R/\mathfrak{J}_{\mathbb{X}}$. As this doubling is an Artinian Gorenstein algebra, we can use its Macaulay inverse system to reduce the LCE problem to a Polynomial Isomorphism (PI) problem for homogeneous polynomials. The last step is polynomial time under some mild assumptions about the codes. Moreover, for indecomposable iso-dual codes we can reduce the LCE search problem to the PI search problem of degree 3 by noting that the corresponding point sets are self-associated and arithmetically Gorenstein, so that we can use the isomorphism problem for the Artinian reductions of the coordinate rings and form their Macaulay inverse systems.

翻译：线性码等价性（LCE）问题被证明等价于点集等价性（PSE）问题，即判断有限域上射影空间中两个点集是否仅相差一个线性坐标变换的问题。对于这样的点集 $\mathbb{X}$，令 $R$ 为其齐次坐标环，$\mathfrak{J}_{\mathbb{X}}$ 为其典范理想。则 LCE 问题可等价转化为其倍环 $R/\mathfrak{J}_{\mathbb{X}}$ 上的代数同构问题。由于该倍环是 Artinian Gorenstein 代数，我们可以利用其 Macaulay 逆系统将 LCE 问题约化为齐次多项式的多项式同构（PI）问题。在关于编码的一些温和假设下，最后一步可在多项式时间内完成。此外，对于不可分解的 iso-dual 码，通过注意到对应的点集是自关联且算术 Gorenstein 的，我们可以将 LCE 搜索问题约化为三次多项式的 PI 搜索问题，从而可利用坐标环的 Artinian 约化的同构问题并构造其 Macaulay 逆系统。