One-Generator Quasi-Cyclic Codes

• One-generator quasi-cyclic codes are principal submodules of a direct sum of cyclic modules that generalize many algebraic error-correcting codes.
• They are constructed using techniques like CRT decomposition and generator matrices, leading to explicit dimensions and BCH-type minimum distance bounds.
• Their algebraic structure supports applications in quantum code design and other advanced coding methods with efficient decoding algorithms.

A one-generator quasi-cyclic (QC) code is a principal submodule of a direct sum of cyclic modules over a commutative ring, and serves as a unifying generalization for numerous families of algebraic error-correcting codes. Over a finite field $\mathbb{F}_q$ (or, more generally, a finite chain ring or local ring), an $\ell$-index QC code of length $n = \ell m$ is an $\mathbb{F}_q$-linear code invariant under cyclic shifts of $\ell$-blocks, admitting an isomorphic module representation as a submodule of $(\mathbb{F}_q[x]/(x^m-1))^\ell$. A one-generator QC code is an $(\mathbb{F}_q[x]/(x^m-1))$-submodule generated by a single element (vector of polynomials). The algebraic and structural theory of such codes provides exact generator/BCH-type bounds, efficient decoding, and applications to quantum codes, as seen in a succession of recent works (Bag et al., 2024, Abdukhalikov et al., 1 Apr 2025, Abdukhalikov et al., 12 Apr 2025, Li et al., 2022, Gao et al., 2013, Wu et al., 2021, Gao et al., 2013, Barbier et al., 2011, Sepasdar, 2017, Lv et al., 2020).

1. Structural Foundations and Algebraic Setup

Let $q$ be a prime power, $m\geq1$, and $\ell\geq2$. The ambient ring is $\ell$0, which is a principal ideal ring when $\ell$1. The ambient vector space $\ell$2 (or $\ell$3) is partitioned into $\ell$4 blocks of length $\ell$5.

A codeword is identified with a vector $\ell$6, where each $\ell$7 is a polynomial of degree less than $\ell$8. A code $\ell$9 is called $n = \ell m$0-quasi-cyclic if it is invariant under the simultaneous action of multiplication by $n = \ell m$1 in each component. This invariance translates to $n = \ell m$2 being an $n = \ell m$3-submodule of $n = \ell m$4. In particular, $n = \ell m$5 is called one-generator if $n = \ell m$6 for some $n = \ell m$7.

When considered over a finite chain ring $n = \ell m$8 (e.g., $n = \ell m$9 with $\mathbb{F}_q$0), or more generally a principal ideal ring, exactly analogous module-theoretic and structural results apply (Gao et al., 2013, Wu et al., 2021, Gao et al., 2013).

2. Construction and Canonical Forms

A one-generator QC code is completely specified by a generator tuple $\mathbb{F}_q$1, and has the form

$\mathbb{F}_q$2

The generator $\mathbb{F}_q$3 often satisfies further divisibility and minimality conditions, often expressed in terms of basic (coprime) decompositions, strong Gröbner bases, or explicit CRT coordinates. In the index-$\mathbb{F}_q$4 case, every such QC code $\mathbb{F}_q$5 is generated by at most two elements and admits a canonical (Gröbner) form (Abdukhalikov et al., 1 Apr 2025, Abdukhalikov et al., 12 Apr 2025):

$\mathbb{F}_q$6

with $\mathbb{F}_q$7 and appropriate compatibility conditions. The code is one-generator if and only if $\mathbb{F}_q$8. In that case, an explicit single generator can be constructed via the extended Euclidean algorithm.

The CRT decomposition of $\mathbb{F}_q$9 allows a direct-sum decomposition of $\ell$0 into constituent codes over extension fields or residue rings, facilitating analysis of dimension, duality, and minimum distance (Abdukhalikov et al., 12 Apr 2025, Gao et al., 2013, Wu et al., 2021).

3. Generator Matrices and Parameter Formulas

The generator matrix of a one-generator QC code is classically constructed by listing the $\ell$1-expansions of the cyclic $\ell$2-shifts of $\ell$3 (Sepasdar, 2017). If $\ell$4, the $\ell$5-dimension is

$\ell$6

Alternatively, for $\ell$7,

$\ell$8

A generator matrix is block circulant: each block corresponding to $\ell$9 gives rise to an $(\mathbb{F}_q[x]/(x^m-1))^\ell$0 circulant matrix, and the rows of the code matrix correspond to shifted versions of $(\mathbb{F}_q[x]/(x^m-1))^\ell$1.

For QC codes over chain rings or principal ideal rings, the full structure is controlled by a sequence of divisors and associated annihilator ideals, linking the $(\mathbb{F}_q[x]/(x^m-1))^\ell$2- or $(\mathbb{F}_q[x]/(x^m-1))^\ell$3-module structure directly to the generator polynomials (Gao et al., 2013, Wu et al., 2021).

4. Duality, Orthogonality, and the LCD Property

The explicit polynomial and matrix representation allows precise duality computations. For the Euclidean, Hermitian, or symplectic inner products, necessary and sufficient polynomial conditions for self-orthogonality or dual containment can be formulated (Bag et al., 2024, Abdukhalikov et al., 1 Apr 2025, Abdukhalikov et al., 12 Apr 2025, Lv et al., 2020). For example, in index-$(\mathbb{F}_q[x]/(x^m-1))^\ell$4,

• Symplectic self-orthogonality: For $(\mathbb{F}_q[x]/(x^m-1))^\ell$5, $(\mathbb{F}_q[x]/(x^m-1))^\ell$6 and $(\mathbb{F}_q[x]/(x^m-1))^\ell$7 circulant, $(\mathbb{F}_q[x]/(x^m-1))^\ell$8 is self-orthogonal iff $(\mathbb{F}_q[x]/(x^m-1))^\ell$9; equivalently,

$(\mathbb{F}_q[x]/(x^m-1))$0

where $(\mathbb{F}_q[x]/(x^m-1))$1 (Bag et al., 2024).

• Euclidean LCD property: $(\mathbb{F}_q[x]/(x^m-1))$2 is LCD iff

$(\mathbb{F}_q[x]/(x^m-1))$3

where $(\mathbb{F}_q[x]/(x^m-1))$4 (Abdukhalikov et al., 12 Apr 2025).

The decomposition via CRT shows that the dual code corresponds to the orthogonal direct sum of constituents, and the LCD property is equivalent to trivial intersection with the dual in each direct summand.

5. Minimum Distance Bounds

Minimum distance lower bounds for one-generator QC codes follow from their CRT decomposition and block cyclic structure. Let $(\mathbb{F}_q[x]/(x^m-1))$5 be the cyclic code of length $(\mathbb{F}_q[x]/(x^m-1))$6 determined by $(\mathbb{F}_q[x]/(x^m-1))$7, and let $(\mathbb{F}_q[x]/(x^m-1))$8 be the code over $(\mathbb{F}_q[x]/(x^m-1))$9 generated by the coefficient vectors of $q$0. Then

$q$1

(Gao et al., 2013). For index-$q$2 QC codes whose cyclic constituent has $q$3 consecutive roots, the BCH-type bound gives $q$4.

Strong bounds also arise for specialized families (quasi-BCH codes, generalized QC codes over $q$5), and combinatorial analysis yields the designed minimum distance in quasi-BCH settings (Barbier et al., 2011).

6. Enumeration, Automorphisms, and Weight Distribution

Enumeration formulas for one-generator QC codes with fixed annihilator are derived from chains of basic divisors and idempotent-lifting algorithms, providing explicit counts depending on the module structure (Gao et al., 2013). For the ambient field case, orbit-counting and group-theoretic analysis of automorphism groups yield tight upper bounds on the number of nonzero codeword weights (Li et al., 2022). The group generated by the block shift, field multipliers, and scalars acts with orbits corresponding to weight levels, and Burnside's lemma gives a closed-form sum:

• $q$6

where $q$7 is the number of nonzero weights, and $q$8 is the automorphism group generated by block shifts and field automorphisms.

7. Applications: Quantum Codes and Extensions

One-generator QC codes underpin constructions of classical codes with strong symplectic or Hermitian self-orthogonality, which are leveraged to construct quantum stabilizer codes and entanglement-assisted codes with record-breaking parameters (Bag et al., 2024, Lv et al., 2020). For example, index-$q$9 QC codes with symplectic self-orthogonality yield quantum codes of parameters $m\geq1$0, where $m\geq1$1 is the symplectic minimum distance.

QC codes over extension rings (such as $m\geq1$2 or finite chain rings) produce families of modules and provide new examples of codes with optimal or near-optimal distance for various metrics, including Lee and Hamming distance (Gao et al., 2013, Wu et al., 2021).

Extensions of the algebraic approach allow for efficient decoding (quasi-BCH syndrome decoding (Barbier et al., 2011)), computation of minimal basis, and explicit generator construction, as well as systematic design of new codes for both classical and quantum applications.

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The theory of one-generator quasi-cyclic codes now forms a central core of algebraic coding theory, linking principal module theory, polynomial factorizations, block-circulant structures, duality, and quantum code construction in a unified algebraic framework (Bag et al., 2024, Abdukhalikov et al., 1 Apr 2025, Abdukhalikov et al., 12 Apr 2025, Li et al., 2022, Gao et al., 2013, Gao et al., 2013, Sepasdar, 2017, Wu et al., 2021, Barbier et al., 2011, Lv et al., 2020).