Dual Codes of Additive Codes

• Dual codes of additive codes are defined via bilinear or character-theoretic orthogonality, forming a vital link between algebraic structure and error-correcting capabilities.
• They employ canonical generator and parity-check matrices to precisely compute parameters across mixed alphabets, cyclic, and chain ring settings.
• Their study yields generalized MacWilliams identities, which enable the translation of weight enumerator relations for practical applications in both classical and quantum codes.

Additive codes are algebraic structures defined as subgroups of an ambient abelian group (often the product of finite fields or rings), with dual codes defined via bilinear or character-theoretic orthogonality. The dual of an additive code is central to its algebraic, combinatorial, and geometric properties. In particular, the study of dual codes of additive codes introduces essential inner products, canonical matrix forms, MacWilliams identities, and connections to complementary duality, cyclic and polycyclic codes, Doob schemes, mixed alphabets, and quantum applications. The following sections systematically present the concept, structure, and analysis of dual additive codes across key settings.

1. Definitions and Bilinear Duality in Additive Code Theory

Let $G$ be a finite abelian group and consider the ambient space $G^n$. An additive code $C \subseteq G^n$ is a subgroup under pointwise addition. The dual code $C^\perp$ is defined with respect to a bilinear pairing, commonly specified as either a canonical inner product (e.g., standard, trace, Hermitian, or hybrid) or via character theory.

• Standard example (mixed ring): For $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$, the canonical inner product is

$$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$

The dual code is $$C^\perp = \{ v \in G^n : \langle u, v \rangle = 0 \quad \forall u \in C \}$$ (0710.1149).

• General abelian setting: For $G$, each $\chi \in \widehat{G}$ is a character, so for $y \in G^n$, the dual under the character table is $G^n$0, where $G^n$1 is the product character associated to $G^n$2 (Agrawal et al., 2023). Symmetric dualities satisfy $G^n$3, but skew-symmetric dualities provide more flexibility, particularly for quantum codes.
• Trace and Hermitian forms (finite fields): For $G^n$4, a trace-based dual is defined as $G^n$5, and similarly for Hermitian variants (Verma et al., 2023, Shi et al., 2022).
• Chain rings: In Eisenstein chain rings, the character-theoretic dual $G^n$6 is defined by vanishing of all additive characters derived from $G^n$7 (Jose et al., 2024).

2. Generator and Parity-Check Matrix Characterizations

For computational and structural analysis, additive codes are described by canonical generator and parity-check matrices, which directly encode duality conditions.

• Z₂Z₄-additive codes: Canonical generator matrix forms allow explicit determination of the dual code's type parameters $G^n$8 via rank conditions and block submatrices. The parity-check matrix for $G^n$9 is constructed so that all generator row pairs are orthogonal under the specified inner product (0710.1149).
• Cyclic and polycyclic settings: For additive cyclic codes over mixed alphabets (e.g., $C \subseteq G^n$0), generator polynomials and their reciprocals, together with Hensel lifts, explicitly yield the dual code's generators (Borges et al., 2014, Roy et al., 2017, Karbaski et al., 2021). In polycyclic codes, Hermitian duals are given by structured shifts and generator polynomials derived as in the cyclic case.
• General matrix criteria: For codes over $C \subseteq G^n$1 or $C \subseteq G^n$2, duality is governed by Gram matrices derived from the generator matrix; invertibility of associated matrices (using entrywise trace or conjugation) is strictly equivalent to the code meeting its dual trivially or being complementary dual—see ACD codes below (Verma et al., 2023, Shi et al., 2022, Benbelkacem et al., 2019).

3. Forms and Parameters of Dual Additive Codes

Distinct additive code settings necessitate specific duality parameter refinements.

• Type parameters (Z₂Z₄): If $C \subseteq G^n$3 has type $C \subseteq G^n$4, then its dual has type $C \subseteq G^n$5 where:

$C \subseteq G^n$6

and $C \subseteq G^n$7 (0710.1149).

• Cyclic codes: For $C \subseteq G^n$8-additive cyclic codes, dual generator polynomials are derived via reciprocal polynomials and appropriate lifts, resulting in direct formulas for all components of the dual code (Borges et al., 2014).
• Doob schemes: Additive duals in the Doob scheme are defined with respect to the trace-based inner product and connect weight/coweight enumerators via MacWilliams transforms. For linear ambient codes, the dual remains code-linear under the same metric; in full generality, duality requires adjustment of the metric (Doob/del Delsarte metric) (Krotov, 2019).
• Homogeneous and Euclidean parameters: In mixed-alphabet and chain ring scenarios, duality maps preserve cardinalities, and generator matrices for duals match the algebraic structure of the ambient ring (Jose et al., 2024).

4. Additive Complementary Dual Codes: Matrix Criteria and Applications

Additive complementary dual (ACD) codes generalize the concept of LCD (linear complementary dual) codes to additive/nonlinear settings. The dual $C \subseteq G^n$9 of an ACD code meets $C^\perp$0 trivially ($C^\perp$1), with generator-matrix invertibility as the decisive criterion.

• Matrix invertibility: For an additive code $C^\perp$2 over $C^\perp$3 or $C^\perp$4, $C^\perp$5 is ACD if and only if the Gram matrix $C^\perp$6 derived (via trace or conjugation) from its generator matrix is invertible over the field (Verma et al., 2023, Benbelkacem et al., 2019, Shi et al., 2022, Ouagagui et al., 7 Aug 2025), and similarly for non-symmetric (skew-symmetric) dualities (Agrawal et al., 2023).
• Gray-maps and binary LCD codes: The image of an additive code under a Gray map (or generalized Gray-like map) can yield binary (or $C^\perp$7-ary) LCD codes; the duality relation between the additive code and its image persists provided orthogonality constraints are maintained (Benbelkacem et al., 2019, Ouagagui et al., 7 Aug 2025).
• Skew-symmetric dualities: When duality is defined by a skew-symmetric matrix (possible only in even dimension over characteristic not 2), the strict invertibility and matrix rank conditions govern the existence and quality of ACD codes, with explicit parameter improvements for certain lengths and minimum distances (Agrawal et al., 2023).

5. MacWilliams Identities for Additive Codes and Their Duals

Duality in additive codes yields generalized MacWilliams identities relating the weight distributions of a code and its dual.

• Doob schemes: In $C^\perp$8, the weight enumerator of $C^\perp$9 is mapped to the coweight enumerator of $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$0 in the dual scheme by

$G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$1

and vice versa, with simplification for linear submodules (Krotov, 2019).

• Gray-mapped codes: For $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$2 additive codes, images under Gray-like maps satisfy

$G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$3

(\emph{MacWilliams-type duality}) (Shi et al., 2018).

• Generalizations: These identities extend directly to multi-level alphabets and to chain ring constructions (Eisenstein codes), maintaining weight/coweight enumerator relations (Jose et al., 2024).

6. Geometric and Quantum Interpretations of Dual Additive Codes

Dual codes of additive codes have direct implications in geometric code theory, quantum error correction, and optimal code construction.

• Subspace packings and faithfulness: In $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$4-additive (QMDS) codes, subspace packing and intersection conditions characterize when both code and dual attain Singleton-optimal parameters (dually QMDS). Faithful packings and geometric quotients provide the necessary and sufficient condition for optimality in both code and its dual (Bartoli et al., 3 Sep 2025).
• Quantum stabilizer codes: For additive codes over finite fields (especially $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$5), the trace Hermitian duality is fundamental for constructing quantum error-correcting codes. Self-orthogonality or ACD property under this form translates into code parameters for entanglement-assisted quantum error correction (Shi et al., 2022).
• Chain ring and mixed alphabet codes: Dual additive codes over Eisenstein chain rings correspond, via a structure-preserving bijection, to Euclidean duals of mixed-alphabet linear codes. This enables efficient code construction, mass-formula computation, and optimal parameter determination under homogeneous or Euclidean metrics (Jose et al., 2024).

7. Examples and Explicit Constructions

The literature provides concrete worked examples illustrating the computation of duals in various additive code contexts.

• Z₂Z₄-additive code example: For $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$6, $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$7 yields $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$8 under the mixed inner product (0710.1149).
• Cyclic code example: For $G = \mathbb{Z}_2^\alpha \times \mathbb{Z}_4^\beta$9, $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$0 with assigned generators, the dual code’s generators are systematically computed via the described reciprocals and Hensel lifting (Borges et al., 2014).
• Polycyclic code example: A right polycyclic code over $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$1 induced by $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$2 is dualized via Hermitian inner product, yielding a sequential code whose generators are derived by reciprocal division of $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$3 (Karbaski et al., 2021).
• Chain ring Eisenstein code example: For $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$4, explicit generators are mapped and the dual characterized via Euclidean duality in the corresponding mixed alphabet code (Jose et al., 2024).
• Doob scheme example: For $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$5, $$\langle u, v \rangle = 2 \sum_{i=1}^\alpha u_i v_i + \sum_{j=1}^\beta u_{\alpha+j} v_{\alpha+j} \pmod{4}$$6 is self-dual, and its weight enumerator matches under the MacWilliams transform (Krotov, 2019).

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The structure and theory of dual codes for additive codes demonstrates broad algebraic and combinatorial richness, enabling rigorous analysis and constructive approaches in both classical and quantum contexts. The canonical forms, bilinear and character-theoretic dualities, and associated matrix criteria provide powerful tools for explicit dual code construction, parameter optimization, and the study of deep properties such as complementary duality, cyclicity, and optimal packing (0710.1149, Borges et al., 2014, Benbelkacem et al., 2019, Verma et al., 2023, Shi et al., 2022, Ouagagui et al., 7 Aug 2025, Agrawal et al., 2023, Bartoli et al., 3 Sep 2025, Krotov, 2019, Jose et al., 2024, Karbaski et al., 2021, Roy et al., 2017, Shi et al., 2018).