Linear Equivalence in Coding Theory

Updated 24 February 2026

Linear equivalence is defined by transformations (monomial, permutation, or semilinear) that preserve key invariants such as Hamming weight and rank metrics.
It facilitates the classification and enumeration of linear, rank-metric, and cyclic codes through projective geometric and combinatorial invariants.
Algorithms reducing equivalence to point-set and graph isomorphism, as well as polynomial isomorphism, inform computational complexity and guide efficient code discovery.

Linear equivalence in coding theory refers to the classification of linear codes up to actions of groups of linear (or monomial/semi-linear) transformations that preserve key code-theoretic invariants—typically the Hamming or rank distance, weight enumerators, and related algebraic or combinatorial properties. The study of linear equivalence has implications for code classification, enumeration, isometry extensions, invariants, and computational complexity. Precise definitions, criteria, enumerative properties, and algorithms have been developed for linear, rank-metric, and additive codes over fields, rings, and various algebraic structures.

1. Notions of Linear Equivalence

The central concept is the equivalence relation on linear codes induced by specific group actions:

Monomial Equivalence (Linear Equivalence): Two codes $C, C' \subseteq \mathbb{F}_q^n$ are monomially equivalent if there exists $M = PD$ (permutation matrix $P$ and invertible diagonal $D$ ) such that $C' = C \cdot M$ . This preserves linearity and the Hamming metric (Kreuzer, 10 Nov 2025, Ball et al., 2021, Dastbasteh et al., 2022).
Permutation Equivalence: Codes are equivalent if related by a coordinate permutation. This is often too coarse, missing structural symmetries induced by scaling.
Semi-linear Equivalence: Over non-prime fields, automorphisms of $\mathbb{F}_q$ may be included, leading to the action of the group $\mathrm{GL}(n,q) \rtimes \mathrm{Aut}(\mathbb{F}_q)$ acting by $v \mapsto \sigma(v)M$ (coordinatewise field automorphism followed by invertible linear map) (Ball et al., 2021, Bouyukliev et al., 2022).
Isometric Equivalence: For linear codes over $\mathbb{F}_q$ , any linear isometry (Hamming-weight preserving linear bijection) is induced by a monomial equivalence—this is a classical result (MacWilliams Extension Theorem) (Greferath et al., 2011, Dastbasteh et al., 2022).

In the context of codes over rings, a monomial map generalizes to allow multiplication by elements of $R^\times$ and permutations, acting componentwise (Greferath et al., 2011).

2. The MacWilliams Extension Theorem and Generalizations

The MacWilliams Extension Theorem is fundamental: any linear weight isometry between linear codes over a finite field (with Hamming weight) extends to a global monomial transformation of the ambient space (Greferath et al., 2011). The theorem has been generalized:

To Codes Over Rings: The question for which weights $w: R \to \mathbb{R}$ (with $w(0) = 0$ ) this extension holds is resolved in terms of "invariant weights"—those with left and right symmetry groups equal to the group of units $R^\times$ (Greferath et al., 2011).
Necessary and Sufficient Conditions: For direct products of chain rings, the extension property holds for $w$ iff certain Möbius-theoretic sums (in terms of weights on principal ideals) are nonzero for all nonzero ideals. For the Hamming and homogeneous weight on Frobenius rings, these conditions are satisfied (Greferath et al., 2011).
Consequences: For weights satisfying the extension theorem, code classification up to equivalence, invariance of weight enumerators, and MacWilliams identities hold. The global orbit-stabilizer principle enables enumeration of equivalence classes.

3. Algorithms and Computational Complexity

Testing linear equivalence (LCE) is critical in classification:

Reduction to Point Set Equivalence: The problem is equivalent to determining if the multisets of column-vectors (projective points in $PG(k-1, q)$ ) differ by a projective linear transformation (Kreuzer, 10 Nov 2025, Bouyukliev et al., 2022). Algorithms exploit projective geometry and combinatorial isomorphism techniques.
Polynomial Isomorphism and Canonical Forms: Recent work reduces LCE to the polynomial isomorphism (PI) problem via Artinian Gorenstein algebra invariants—specifically Macaulay inverse systems. For indecomposable iso-dual codes, the PI problem reduces to isomorphism of cubic forms (IP1S) (Kreuzer, 10 Nov 2025).
Graph-Based Invariants (ELC): For binary codes, edge-local complementation on bipartite graphs precisely captures code equivalence; the ELC-orbit corresponds to the code equivalence class. Efficient classification algorithms iterate through ELC-moves (0710.2243).
Rank-Metric Codes: For matrix and rank-metric codes, code equivalence (matrix code equivalence, MCE) is at least as hard as the classical monomial equivalence problem for Hamming metric, and thus as hard as graph isomorphism in the worst case (Couvreur et al., 2020). For $\mathbb{F}_{q^m}$ -linear codes, equivalence reduces to right-equivalence of matrix codes, which can be decided in deterministic polynomial time or ZPP depending on the field size (Couvreur et al., 2020).

4. Enumeration and Asymptotics

Classification up to equivalence has been quantitatively analyzed:

Counting Classes: For a fixed field size $q$ and length $n$ , the number of equivalence classes of $k$ -dimensional codes under monomial or semilinear equivalence admits precise asymptotics (Giusto et al., 16 Oct 2025):

$N_{k, n}^{M} \sim \frac{q^{k(n-k)}}{K_q n!(q - 1)^{n-1}}$

where $K_q = \prod_{j=1}^\infty (1-q^{-j})$ . The distribution of dimensions among equivalence classes is asymptotically discrete Gaussian centered at $n/2$ in the large $n$ limit.

Rank-Metric Codes: For Gabidulin and twisted Gabidulin codes, the number of inequivalent codes scales as products of $(q^m - q^i)$ , with finer lower and upper bounds available, and explicit connections to invariants derived from Galois automorphism sequences (Neri et al., 2019).

5. Structural and Invariant Properties

Equivalence characterizations induce effective invariants:

Projective Geometric Signatures: Equivalence coincides with projective equivalence of the set of column points in the projective space, reducing the problem to incidence and structure of multisets in $PG(k-1, q)$ (Bouyukliev et al., 2022, Kreuzer, 10 Nov 2025).
Rank-Metric Isometry Invariants: In the rank metric, equivalence classes are determined by the sequence of subspace dimensions generated by successive Galois automorphisms and their intersections, leading to computable invariants for MRD codes (Neri et al., 2019).
Family-Specific Invariants: For cyclic and constacyclic codes, algebraic criteria involving shifts, multipliers, and affine actions on defining sets offer sharp equivalence tests, facilitating search for new codes (Dastbasteh et al., 2022).
Extension Property and Duality: Once the extension property holds (i.e., linear isometries extend globally), usual duality and MacWilliams identities apply, and code classification reduces to group actions (Greferath et al., 2011).

6. Applications and Implications

Understanding linear equivalence has direct impact:

Code Classification: Enables complete and non-redundant catalogs of linear codes, elimination of duplicates in exhaustive computer searches, and systematic exploration of new code families (Dastbasteh et al., 2022, 0710.2243).
Complexity of Isometry Testing: The equivalence problem is GI-hard in the general case, with practical polynomial-time algorithms for many important subclasses (e.g., $\mathbb{F}_{q^m}$ -linear rank-metric codes) and is deeply connected to isomorphism problems in algebraic combinatorics (Kreuzer, 10 Nov 2025, Couvreur et al., 2020).
Unified Analytical Techniques: Recent advances show that thresholds for combinatorial properties such as list-decoding and list-recovery transfer uniformly across random codes and code families whose local combinatorial structure can be equivalently analyzed, as in locality equivalence of RLCs and RS codes (Levi et al., 2024).
Duality and Invariants: The establishment of MacWilliams-type identities, dual code properties, and associated combinatorial invariants follow from extension theorems for invariant weights (Greferath et al., 2011).

7. Special Topics: Cyclic, Constacyclic, and Rank-Metric Codes

Cyclic and Constacyclic Codes: Equivalence is established via shifts, multipliers, and affine maps acting on the defining set of code zeros, with sharp group-theoretic criteria and impact on efficient code search (Dastbasteh et al., 2022).
Rank-Metric Codes: Equivalence is determined through action of linear and semilinear groups incorporating $\mathbb{F}_{q^m}$ -automorphisms and $GL_n(q)$ , with extension to classification and count of inequivalent Gabidulin and twisted Gabidulin codes (Neri et al., 2019, Gupta et al., 2022).
Edge Local Complementation: For binary linear codes, the ELC-orbit of a graph is a complete invariant for code equivalence; its size enumerates the number of information sets, and minimal vertex degrees determine minimal code distance (0710.2243).