Orthogonal Self-Dual Codes

• Orthogonal Self-Dual Codes are linear codes defined over finite fields, rings, or group algebras that equal their duals under a prescribed symmetric bilinear form.
• Key methodologies include GRS constructions, cyclotomic coset techniques, and algebraic lifting methods to achieve MDS, near-MDS, and almost self-dual constructions.
• These codes have significant applications in quantum error correction, lattice theory, and invariant theory, supported by explicit enumeration and asymptotic growth results.

Orthogonal self-dual codes are linear codes defined over finite fields, rings, or group algebras that are equal to their own duals under a prescribed symmetric bilinear form, most commonly the Euclidean inner product. These codes are central in coding theory for their algebraic structure, optimality properties (such as saturating the Singleton bound in the MDS context), and intrinsic connections to lattices, combinatorial designs, and quantum error correction.

1. Definitions and Fundamental Constructions

Let $\mathbb{F}_q$ be a finite field of characteristic $p$, $n$ a positive integer, and let $C \subseteq \mathbb{F}_q^n$ be a linear code of length $n$ and dimension $k$. The Euclidean inner product on $\mathbb{F}_q^n$ is

$\langle x, y \rangle = \sum_{i=1}^n x_i y_i.$

The Euclidean dual code is $$C^\perp = \{ y \in \mathbb{F}_q^n \mid \langle x, y \rangle = 0,\, \forall x \in C \}$$.

• $C$ is self-orthogonal if $p$0.
• $p$1 is self-dual if $p$2 (then $p$3 and $p$4 even).
• Orthogonal here refers to the code being isotropic under the symmetric form; all codewords are orthogonal to each other.

A code is maximum distance separable (MDS) if it attains the Singleton bound: $p$5. A code is almost self-dual if $p$6 is odd, $p$7 and $p$8 (Fang et al., 2019).

Extensions to codes over rings, group algebras, or chain rings are analogously defined with respect to suitable inner products.

2. Algebraic Criteria for (Self-)Orthogonality in MDS and GRS Constructions

A principal approach to constructing orthogonal self-dual codes uses Generalized Reed–Solomon (GRS) codes and their extensions. The central criterion is as follows (Fang et al., 2019):

• For $p$9 over $n$0 with distinct evaluation points $n$1 and nonzero multipliers $n$2,

$n$3

where $n$4.

In the special case $n$5 over square fields $n$6, the criterion simplifies: if there exists $n$7 such that $n$8 is a nonzero square for all $n$9, then $C \subseteq \mathbb{F}_q^n$0 is self-dual. Using cyclotomic coset-based evaluation sets, about $C \subseteq \mathbb{F}_q^n$1 new MDS self-dual codes can be obtained for each large square $C \subseteq \mathbb{F}_q^n$2 (Fang et al., 2019).

Analogous orthogonality conditions are available for

• Extended GRS codes (for almost self-dual and self-dual codes with odd $C \subseteq \mathbb{F}_q^n$3).
• Twisted GRS (TGRS) codes, where explicit block-matrix conditions control multi-twist orthogonality and self-duality for a large new class of MDS and near-MDS codes (Chen et al., 22 May 2026).

For codes constructed as $C \subseteq \mathbb{F}_q^n$4-quasi-abelian group algebra submodules (e.g., double circulant codes), self-duality reduces to the equation $C \subseteq \mathbb{F}_q^n$5 in the group ring, which has solutions if and only if $C \subseteq \mathbb{F}_q^n$6 is a square in the ambient field (Lin et al., 2021).

3. Infinite Families, Enumeration, and Asymptotic Results

Enumeration and asymptotic properties are key for evaluating the abundance and quality of orthogonal self-dual codes.

• MDS self-orthogonal and self-dual codes: The cyclotomic coset construction for GRS codes yields about $C \subseteq \mathbb{F}_q^n$7 distinct even lengths $C \subseteq \mathbb{F}_q^n$8 with true self-dual codes, $C \subseteq \mathbb{F}_q^n$9 with almost self-dual codes (odd $n$0), and $n$1 with self-orthogonal codes of dimension $n$2 (even $n$3), demonstrating quadratic growth in the number of distinct parameters as $n$4 grows (Fang et al., 2019).
• Quasi-abelian index-2 codes: The counting formulas for the number of self-dual and self-orthogonal index-2 quasi-abelian codes are explicit and depend on the number of solutions to $n$5 in field components; the existence of self-dual codes hinges on $n$6 being a square (Lin et al., 2021).
• Chain ring and ring codes: Recursive constructions yield explicit enumeration formulas for all self-orthogonal and self-dual codes over arbitrary finite commutative chain rings of even characteristic, expressed in terms of chains of self-orthogonal codes over their residue fields (Yadav et al., 7 Oct 2025, Yadav et al., 7 Oct 2025).

Orthogonal self-dual code families built via these methods are asymptotically good—there exist infinite sequences with rate $n$7 and lower-bounded relative distance, and for certain parameters they exceed classical bounds (e.g., surpassing the Gilbert–Varshamov bound for nonprime $n$8) (Bassa et al., 2017).

4. Recursive and Algebraic Lifting Methods

Orthogonal self-dual codes can be systematically constructed from more elementary self-orthogonal codes:

• Binary case: Every self-orthogonal code of even length (with further divisibility constraint if $n$9) extends to a self-dual code (Bassa et al., 2017, An et al., 7 Nov 2025). This is achieved via Witt-type isometries or algebraic matrix completion processes.
• Ring and chain ring case: Any self-orthogonal code over a finite chain ring can be recursively lifted from a chain of self-orthogonal residue codes (Yadav et al., 7 Oct 2025, Yadav et al., 7 Oct 2025). These liftings preserve orthogonality and the chain structure of code invariants (torsion codes, types).
• $k$0 case: Any self-orthogonal $k$1-code can be expanded to a self-dual code by adjoining vectors of the form $k$2 for certain $k$3 in the binary dual or via block generator augmentation. This method recovers all known indecomposable self-dual codes of small length and has led to the discovery of new Euclidean-optimal codes and extremal unimodular lattices (Shi et al., 2024).
• $k$4 case: All code types (I, II, 0) can be realized, with explicit canonical generator block forms, product, and neighbor constructions (0910.3084).

5. Applications, Structure Theorems, and Quantum Codes

• Quantum error correction: Any classical self-orthogonal code under the (Euclidean or Hermitian) inner product can be used to construct a quantum stabilizer code via the CSS construction. Many families of self-orthogonal and self-dual MDS or near-MDS codes yield quantum codes saturating the quantum Singleton bound (Wang et al., 1 Jun 2025, Wang et al., 14 Jan 2025, Chen et al., 22 May 2026).
• Algebraic–Geometry (AG) codes: Orthogonality and self-duality for AG codes is characterized by differential (residue) criteria; generically, an AG code $k$5 is self-orthogonal if $k$6 and self-dual if $k$7 (with $k$8 a canonical divisor) (Wang et al., 14 Jan 2025, Wang et al., 1 Jun 2025).
• Lattice theory: Via Construction A, self-dual codes over $k$9 map to (odd or even) unimodular lattices; new extremal lattices have been constructed in this way from self-dual codes of previously unknown parameters (Shi et al., 2024).
• Invariant theory: Weight enumerators for self-dual codes are governed by explicit rings of invariants and MacWilliams-type relations, depending on code type and choice of module (e.g., for $\mathbb{F}_q^n$0-codes, the invariance group is $\mathbb{F}_q^n$1) (0910.3084).

6. Broadening Constructions: Plateaux Functions, Embedding, and Algebraic Techniques

• Plateaued functions: By using weakly regular plateaued functions (a generalization of bent functions), large families of $\mathbb{F}_q^n$2-ary linear self-orthogonal and self-dual codes with controlled weight spectra are constructed, particularly optimal or almost optimal for binary and ternary cases (Wang et al., 2024).
• Shortest embedding: Minimal self-orthogonal or self-dual extensions of arbitrary linear codes can be computed via properties of the hull and orthonormal basis construction. For Hamming and Reed–Muller codes, this yields, for example, the shortened Golay code and previously unknown optimal self-orthogonal codes (An et al., 7 Nov 2025).
• Orthogonal matrices: In characteristic $\mathbb{F}_q^n$3, randomized sampling in the orthogonal group $\mathbb{F}_q^n$4 enables direct construction of MDS self-dual codes using canonical generator forms $\mathbb{F}_q^n$5 with $\mathbb{F}_q^n$6 (Shi et al., 2016).

7. Comparison with Previous Literature and Impact

Recent advances have dramatically increased the known set of orthogonal self-dual and self-orthogonal codes, particularly in the MDS and AG setting. Where classical constructions (e.g., via cyclic, double-circulant, or direct-sum methods) were limited to $\mathbb{F}_q^n$7 or $\mathbb{F}_q^n$8 possible parameters, cyclotomic and group algebra methods expand this to $\mathbb{F}_q^n$9 or $\langle x, y \rangle = \sum_{i=1}^n x_i y_i.$0. These growth rates are confirmed through explicit counts and constructive enumeration theorems (Fang et al., 2019, Lin et al., 2021).

Orthogonal self-dual codes thus provide a unifying algebraic framework encompassing many optimal and structurally rich code families, with deep connections to algebraic geometry, finite group theory, combinatorics, and the theory of quantum information (Fang et al., 2019, Lin et al., 2021, Wang et al., 1 Jun 2025, Bassa et al., 2017, Yadav et al., 7 Oct 2025, Shi et al., 2016, Wang et al., 2024, Shi et al., 2024, Wang et al., 14 Jan 2025, 0910.3084, An et al., 7 Nov 2025, Chen et al., 22 May 2026).