Linear Code Equivalence Problem

• Linear Code Equivalence is the problem of determining if two k-dimensional codes over finite fields are equivalent under coordinate permutations, nonzero scalar multiplications, and field automorphisms.
• The topic employs algebraic, geometric, and combinatorial reductions, including transformations to Graph Isomorphism and polynomial isomorphism problems, to reveal its computational complexity.
• Its applications in code classification and cryptanalysis highlight critical impacts on cryptographic security and the theoretical limits of algorithmic efficiency.

Linear Code Equivalence (LCE) is the problem of deciding whether two linear codes, typically represented as $k$-dimensional subspaces of $F^n$ over a finite field $F$, are equivalent under a well-characterized transformation group. This transformation usually includes coordinate permutations, multiplications by nonzero scalars (monomial transformations), and potentially field automorphisms. LCE sits at the intersection of algebraic coding theory, combinatorics, group theory, and computational complexity, underlying key problems in code classification, cryptographic security, and algorithmic isomorphism theory.

1. Definitions and Variants

Let $F$ be a finite field of order $q$, and let $A, B \subseteq F^n$ be $k$-dimensional linear codes. The most commonly studied forms of code equivalence are:

1. Permutation Code Equivalence (PCE): $A$ and $B$ are permutation equivalent if there exists $\pi \in S_n$ such that $$B = \{(u_{\pi(1)},...,u_{\pi(n)}) : (u_1,...,u_n) \in A\}$$. Matrix form: if $G_A$ and $G_B$ are generator matrices, then $G_B = S G_A X$ for some $S \in GL_k(F)$ and permutation matrix $X$.
2. Linear Code Equivalence (LCE): Allows permutations, multiplications by nonzero scalars in each coordinate (monomial transformations), and in some formulations field automorphisms. Two codes $A, B$ are LCE-equivalent if there exists a monomial matrix $M$ and $X \in GL_k(F)$ such that $G_B = X G_A M$.
3. Semi-linear equivalence: Involves a global field automorphism $\sigma \in \mathrm{Aut}(F)$ in addition to monomial actions. $A$ and $B$ are semi-linearly equivalent if $$u = (u_1,...,u_n) \mapsto (a_1 \sigma(u_{\pi^{-1}(1)}),...,a_n \sigma(u_{\pi^{-1}(n)}))$$ for some $\pi \in S_n$, $a_i \in F^*$.

These forms coincide with the isometry (Hamming-distance-preserving) group for linear codes over finite fields.

2. Algebraic, Geometric, and Group-Theoretic Reductions

Algebraic Structure and Equivalence Group

The LCE group over $F^n$ is the semi-direct product $\mathrm{Mon}_n(F) \rtimes \mathrm{Aut}(F)$, where $\mathrm{Mon}_n(F)$ is the monomial matrices (coordinate permutations and coordinate-wise scaling). For codes over rings, every linear isometry extends to a monomial transformation provided the weight function satisfies the Extension Theorem (Greferath et al., 2011). The group-theoretic structure clarifies that the complexity of LCE is bounded above by the combinatorics of these transformations, rather than exponential in $q^n$.

Connections to Graph Isomorphism

For codes with trivial hulls (i.e., $A \cap A^{\perp} = \{0\}$), deciding permutation equivalence is deterministically polynomial-time reducible to Graph Isomorphism (GI) (Bardet et al., 2019). Given generator matrix $G_A$, compute the projection matrix $P_A = G_A^{T} (G_A G_A^{T})^{-1} G_A$; associate to $A$ a weighted graph with adjacency matrix $P_A$. Then $A$ and $B$ are permutation equivalent iff their graphs are isomorphic. This reduction runs in $O(n^\omega + n^2 + T_{GI}(n))$ time (where $\omega$ is the matrix multiplication exponent). Experimental results show codes of length $5\times10^4$ can be handled in minutes with state-of-the-art GI solvers.

Projective, Geometric, and Polynomial Reductions

The LCE problem can be reformulated as the Point Set Equivalence (PSE) problem in projective space (Kreuzer, 10 Nov 2025). A generator matrix with nonzero, non-repeated columns up to scale gives rise to a point set $X$ in projective space $\mathbb{P}^{k-1}(F)$. LCE then becomes PSE: are $X$ and $Y$ equivalent under $GL_k(F)$? This is further reduced: the coordinate ring $R_X$ of $X$ leads to Artinian Gorenstein algebra $D_X = R_X / J_X$ (with $J_X$ the canonical ideal) whose Macaulay inverse system directly encodes the equivalence. Finally, LCE is polynomial-time reducible to a polynomial isomorphism (PI) problem: decide if there is $A \in GL_k(F)$ with $f(A^{-1}x) = g(x)$ for the associated polynomials $f, g$.

Reductions among Equivalence Problems

There exists a polynomial-time Karp reduction from PCE to LCE (Cheraghchi et al., 11 Feb 2025). Enlarging the dimension by one and concatenating generator matrices (and corresponding block structures), one can embed any instance of PCE (permutation only) in LCE (monomial). The hardness equivalence among PCE, LCE, Graph Isomorphism (GI), and Lattice Isomorphism (LIP) places these in the "intermediate" complexity tier.

3. Complexity and Algorithmic Frameworks

Quasipolynomial & Polynomial Reductions

Babai's quasipolynomial GI algorithm provides the best-known upper bound for trivial-hull (GI-reducible) instances. For general LCE, especially over fields with nontrivial hull, code shortening or change of inner product (Hermitian over $F_{p^m}$) is required; complexity may rise to $O(h n^{\omega+h+1} T_{GI}(n))$ when the hull dimension is $h$ (Bardet et al., 2019). For "good" codes (high rate, sufficient minimum distance), reductions via Macaulay inverse systems are polynomial time in $n,k,\log q$ (Kreuzer, 10 Nov 2025).

Geometric and Combinatorial Algorithms

LCE is also naturally reduced to binary matrix isomorphism using geometric incidence (Bouyukliev et al., 2022). The CEIMPG algorithm constructs the full incidence matrix of hyperplane-point relations in projective space and appends the code's characteristic vector. CESIMPG, optimized for prime fields, shortens this to a much smaller matrix associated only to those points present in the code. Both methods reduce LCE to colored matrix isomorphism and solving small linear systems for the required scaling and automorphism. Prior to these, Leon's algorithm operates on weight enumerators of codeword orbits.

Graph-Theoretic Algorithms (Binary Codes)

For binary codes, a strong combinatorial approach uses edge local complementation (ELC) on bipartite graphs corresponding to standard-form generator matrices (0710.2243). The ELC orbit of a bipartite graph parametrizes the equivalence class of codes. Information sets and minimum distance correspond to structural features of the orbit. The algorithm is competitive up to length $n=16$ (days on a small cluster), and ELC orbits provide canonical representatives.

4. Structural Results and Theorems

MacWilliams Extension Theorem and Semi-Linear Equivalence

Every isometry between linear codes over finite fields is induced by a monomial transformation and, if the field has automorphisms, a global field automorphism (Ball et al., 2021). Thus, LCE reduces the search space dramatically from $(q!)^n n!$ for arbitrary permutations to $h n! (q-1)^n$ for monomial $\rtimes$ automorphism actions, where $h = \log_p q$. For additive MDS codes, additive equivalence arises analogously.

Equivalence Criteria for Specific Code Families

For cyclic and constacyclic codes, necessary and sufficient conditions for monomial equivalence are given by shift maps and affine actions on the defining set (Dastbasteh et al., 2022). For constacyclic codes over $\mathbb{F}_4$ with $\gcd(3n, \varphi(3n))=1$, all permutation equivalence arises from the action of multipliers, yielding a complete characterization and facilitating search pruning.

Algebraic Conditions over Rings

The Extension Theorem holds for LCE over finite rings $R$ if the weight $w$ satisfies a non-vanishing criterion involving Möbius inversion and principal ideal lattices (Greferath et al., 2011). For each $x$ in the index set, the diagonal sum $$D_x = \sum_{x^\perp \le t} \mu(0, Rx t) w(t x) \neq 0$$ ensures every isometry between $R$-linear codes extends to a monomial transformation.

5. Applications, Cryptography, and Limitations

Cryptanalysis and Hidden Subgroup Problem

LCE and PCE are central to the security of code-based cryptosystems (e.g., McEliece, Sidelnikov). The LCE problem reduces to a nonabelian hidden subgroup problem (HSP), suggesting potential quantum algorithms. However, for widely-used codes (Goppa, Reed-Muller), quantum Fourier sampling is ineffective unless highly entangled measurements are available (Dinh et al., 2011). Classical attacks like Sendrier’s Support Splitting Algorithm (SSA) break McEliece for Goppa codes in polynomial time if the code is known. Sidelnikov’s system based on RM codes is quantum-resistant by HSP-hardness, but low-rate cases are vulnerable to quasipolynomial classical attacks.

Limitations and Open Problems

For codes with nontrivial hull, GI equivalence fails, necessitating code shortening or different inner products. Extending reductions to ring codes or lattices over arbitrary fields and composite rings remains open. There is also an unresolved gap between quasipolynomial and polynomial time for general LCE. Improved reductions, search-to-decision equivalence, and better algorithms for large parameters are active research directions.

6. Classification and Representative Examples

Recent work provides explicit classification results and enumeration formulas for equivalence classes in specialized code families—e.g., the family of $2$-dimensional MRD codes over $F_{q^{2t}}$ (Gupta et al., 2022). Four congruence conditions on the shift and scalar parameters, incorporating Galois automorphism actions, fully determine equivalence; the enumeration is exact for $t \geq 5$ and carries over to $t=4$ by refined arguments.

7. Summary Table: Algebraic Reduction Complexity

Reduction Type	General Instance	Trivial-Hull Instance	Binary Case (ELC Orbit)
Algebraic (GI)	$O(h n^{\omega+h+1} T_{GI}(n))$	$O(n^\omega + n^2 + T_{GI}(n))$	$O(N_n\, \mathrm{poly}(n))$
Polynomial Isomorphism (PI)	$\mathrm{poly}(n, k, \log q)$	--	--
Matrix Isomorphism (incidence/CESIMPG)	$O(T_{iso}(r,n) + k^3 + kn)$	--	--

The table summarizes principal reduction complexities for deciding LCE across principal structural cases as established in (Bardet et al., 2019, Kreuzer, 10 Nov 2025, Bouyukliev et al., 2022), and (0710.2243).

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The paper of Linear Code Equivalence encompasses deep results in algebraic, geometric, combinatorial, and complexity-theoretic domains. The tight computational and structural relationship to Graph Isomorphism, polynomial isomorphism, and group action counts underpins both effective algorithms for code classification and foundational hardness in code-based cryptography. The field continues to address open problems in efficiency, broader algebraic structures, and the quantum attack surface.