Code Equivalence, Point Set Equivalence, and Polynomial Isomorphism

Abstract: 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.