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Code Equivalence, Point Set Equivalence, and Polynomial Isomorphism

Martin Kreuzer

TL;DR

The paper proves that the Linear Code Equivalence (LCE) problem is polynomial-time equivalent to the Point Set Equivalence (PSE) problem by interpreting generator matrices as sets of points in ${\mathbb{P}}^{k-1}$ and employing the canonical module and the doubling construction to form an Artinian Gorenstein algebra. Through the Macaulay inverse system, the LCE/PSE problems are reduced to a Polynomial Isomorphism (PI) problem for homogeneous polynomials of degree $2r_{\mathbb X}-1$, with polynomial-time reductions under mild code assumptions. For indecomposable iso-dual codes, the corresponding point sets are self-associated, yielding a further reduction to PI in degree 3 (the cubic equivalence problem, IP1S). This work links coding theory to classical isomorphism problems in algebraic geometry and cryptography, providing a framework for analyzing the computational hardness of LCE and related problems and guiding practical implementations of the reductions.

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.

Code Equivalence, Point Set Equivalence, and Polynomial Isomorphism

TL;DR

The paper proves that the Linear Code Equivalence (LCE) problem is polynomial-time equivalent to the Point Set Equivalence (PSE) problem by interpreting generator matrices as sets of points in and employing the canonical module and the doubling construction to form an Artinian Gorenstein algebra. Through the Macaulay inverse system, the LCE/PSE problems are reduced to a Polynomial Isomorphism (PI) problem for homogeneous polynomials of degree , with polynomial-time reductions under mild code assumptions. For indecomposable iso-dual codes, the corresponding point sets are self-associated, yielding a further reduction to PI in degree 3 (the cubic equivalence problem, IP1S). This work links coding theory to classical isomorphism problems in algebraic geometry and cryptography, providing a framework for analyzing the computational hardness of LCE and related problems and guiding practical implementations of the reductions.

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 , let be its homogeneous coordinate ring and its canonical ideal. Then the LCE problem is shown to be equivalent to an algebra isomorphism problem for the doubling . 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.

Paper Structure

This paper contains 10 sections, 24 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Linear Code Equivalence and Point Set Equivalence
  3. The Canonical Module
  4. Iso-Dual Codes and Self-Associated Sets of Points
  5. The Canonical Ideal
  6. Doublings of Sets of Points
  7. The Macaulay Inverse System
  8. Point Set Equivalence and Polynomial Isomorphism
  9. The Linear Code Equivalence Problem for Iso-Dual Codes
  10. Conclusion and Outlook

Key Result

Theorem 2.1

$\mathstrut$ Let $C,C'$ be two linear $[n,k]_q$-codes. Then $C$ and $C'$ are (linearly) isometric if and only if there exist a permutation matrix $P \in \mathop{\rm GL}\nolimits_n({\mathbb{F}}_q)$ and an invertible diagonal matrix $D\in \mathop{\rm GL}\nolimits_n({\mathbb{F}}_q)$ such that $C' = \{

Theorems & Definitions (59)

  • Theorem 2.1: MacWilliams Extension Theorem
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6: Linear Code Equivalence and Point Set Equivalence
  • proof
  • Remark 2.7
  • Definition 2.8
  • ...and 49 more