1 Introduction↩︎

Quantum error-correcting codes (QECCs) are essential for protecting quantum information from decoherence and operational errors, and therefore play a central role in the realization of large-scale quantum computation and reliable quantum communication. The theory of QECCs has developed rapidly since the seminal works of Shor [1] and Steane [2]. The stabilizer formalism, established by Calderbank, Rains, Shor, and Sloane [3], provides a powerful bridge between quantum codes and classical codes over finite fields. In particular, binary stabilizer codes can be obtained from classical quaternary linear codes that are self-orthogonal with respect to the Hermitian inner product. This connection was extended to the nonbinary setting in [4]–[6], leading to the Hermitian construction of \(q\)-ary quantum stabilizer codes from Hermitian self-orthogonal codes over \(\mathbb{F}_{q^2}\).

The Hermitian construction requires the classical ingredient code \(\mathcal{C}\) to satisfy the dual-containing condition \(\mathcal{C}^{\perp_h} \subseteq \mathcal{C}\), or equivalently, to be Hermitian self-orthogonal. Many elegant families of quantum codes have been obtained from algebraic codes that meet this requirement, including cyclic, constacyclic, and quasi-cyclic codes [7]–[19]. However, the dual-containing condition is rather restrictive and excludes many good classical codes from being used directly. Consequently, relaxing this condition has been an important research direction in quantum coding theory.

A significant step in this direction was taken by Lisoněk and Singh [20], who proposed a quantum version of Construction X that produces binary stabilizer codes from arbitrary quaternary linear codes, provided that the code is “nearly” self-orthogonal, i.e., the dimension of the quotient \(\mathcal{C}^{\perp_h}/(\mathcal{C} \cap \mathcal{C}^{\perp_h})\) is positive but small. Their idea was later extended to general finite fields and refined by Degwekar et al. [21], Ezerman et al. [22], [23], and Hu and Liu [24]. In a related direction, Wang et al. [25] introduced a dual-extension method that constructs a Hermitian dual-containing code from an arbitrary quaternary linear code by appending \(r\) redundant coordinates, where \(r=\mathop{\mathrm{Rank}}(HH^\dagger)\) is the rank of the Hermitian inner-product matrix of a parity-check matrix \(H\). Explicit constructions from quasi-cyclic codes were further investigated by Lv et al. [26], who derived stabilizer codes with good parameters by exploiting the symplectic dual structure of QC codes.

Despite these advances, most existing extension methods either rely on specific algebraic structures of the underlying classical codes or require ad hoc choices of extension vectors. A unified and explicit framework that works for arbitrary linear codes over \(\mathbb{F}_{q^2}\) and systematically selects extension vectors is still desirable. In this paper, we propose such a framework by using the Jordan canonical form of the Hermitian inner-product matrix \(A = HH^\dagger\). The Jordan decomposition \(A = P J_A P^{-1}\) provides a natural set of extension vectors, namely the columns of \(P^{-1}\), which form a Jordan basis of \(A\). By successively reducing the rank of \(A\) via rank-one perturbations along this basis, we construct a Hermitian self-orthogonal code \(\mathcal{C}_{\mathrm{so}} = [n+r, n-k]_{q^2}\) from any \(r\)-nearly self-orthogonal code \(\mathcal{C} = [n,k,d]_{q^2}\), where \(r = \mathop{\mathrm{Rank}}(HH^\dagger) = (n-k) - \dim_{\mathbb{F}_{q^2}}(\mathcal{C}^{\perp_h}\cap \mathcal{C})\). Applying the Hermitian construction then yields a \(q\)-ary quantum code with parameters \([[n+r, 2k-n+r, \geq d]]_q\). This approach is systematic and does not require the classical code to possess any special algebraic structure beyond linearity.

The remainder of this paper is organized as follows. Section II reviews the necessary notation and preliminary results on Hermitian matrices and linear codes over finite fields. Section III presents the general extending construction based on the Jordan canonical form. Section IV provides several examples of new quantum codes. Section V concludes the paper.

2 PRELIMINARIES AND NOTATION↩︎

We first introduce notation and fundamental properties of Hermitian matrices, cyclic codes, quasi-cyclic codes, and linear codes over finite fields[27]–[30].

Let \(q\) be a prime power, and \(\mathbb{F}_q\) denote the finite field with \(q\) elements. The degree-2 extension field \(\mathbb{F}_{q^2}\) is equipped with a Hermitian conjugate operation induced by the Frobenius automorphism:

• For any element \(x \in \mathbb{F}_{q^2}\), its Hermitian conjugate is defined as \(x^\dagger = x^q\). An element satisfies \(x^\dagger = x\) if and only if \(x \in \mathbb{F}_q\).

• For any matrix \(M = (m_{ij}) \in \mathbb{F}_{q^2}^{m \times n}\), its Hermitian conjugate transpose is \(M^\dagger = (m_{ji}^\dagger) \in \mathbb{F}_{q^2}^{n \times m}\). A square matrix \(A \in \mathbb{F}_{q^2}^{m \times m}\) is Hermitian if \(A^\dagger = A\).

A core property of Hermitian matrices over \(\mathbb{F}_{q^2}\) is that all eigenvalues lie in \(\mathbb{F}_q\) (since any eigenvalue \(\lambda\) satisfies \(\lambda^q = \lambda\) from the Hermitian constraint). Thus, the Jordan canonical form \(J_A\) of a Hermitian matrix always exists over \(\mathbb{F}_{q^2}\), with all diagonal entries in \(\mathbb{F}_q\).

The norm map \(N: \mathbb{F}_{q^2} \to \mathbb{F}_q\), a critical tool for constructing rank-1 perturbations, is defined as: \[N(x) = x \cdot x^\dagger = x^{q+1}, \quad \forall x \in \mathbb{F}_{q^2}.\] This map is a surjective group homomorphism from the multiplicative group \(\mathbb{F}_{q^2}^*\) to \(\mathbb{F}_q^*\): for any target value \(c \in \mathbb{F}_q^*\), there exists \(t \in \mathbb{F}_{q^2}^*\) such that \(N(t) = c\).

For \(M\) over \(\mathbb{F}_{q^2}\), we denote \(\mathop{\mathrm{Rank}}(M)\) as its rank, \(\mathop{\mathrm{Im}}(M)\) as its column space (range), and \(\mathop{\mathrm{Ker}}(M)\) as its null space (kernel). We use two standard results throughout the proof:

1. Sylvester’s rank inequality for rank-1 perturbations: For any matrix \(M \in \mathbb{F}_{q^2}^{m \times m}\) and non-zero vector \(\boldsymbol{u} \in \mathbb{F}_{q^2}^m\), \[\mathop{\mathrm{Rank}}(M) - 1 \leq \mathop{\mathrm{Rank}}(M + \boldsymbol{u}\boldsymbol{u}^\dagger) \leq \mathop{\mathrm{Rank}}(M) + 1.\]

2. Jordan basis property: If \(A = P J_A P^{-1}\) where \(J_A\) is the Jordan canonical form of \(A\), the columns of \(P^{-1}\) form a complete Jordan basis of \(A\), consisting of eigenvectors and generalized eigenvectors spanning \(\mathbb{F}_{q^2}^m\).

A classical linear code \(\mathcal{C}\) of length \(n\) over \(\mathbb{F}_{q^{2}}\) is a non-empty subset of \(\mathbb{F}_{q^{2}}\), denoted as \([n,k,d]_{q^{2}}\). Suppose that \(\vec{u}=(u_{0},\ldots,u_{n-1})\) and \(\vec{v}=(v_{0},\ldots,v_{n-1})\) are vectors in \(\mathbb{F}_{q^{2}}\). We define the Hermitian inner product as \(\langle\vec{u}, \vec{v}\rangle_{h}=\sum_{i=0}^{n-1} u_{i} v^q_{i}\). The weight of \(\vec{u}\) is the number of nonzero coordinates in \(\vec{u}\), which is denoted by \(wt(\vec{u})\). The minimum non-zero Hamming weight of \(\mathcal{C}\) is \(d(\mathcal{C})=\min \left\{wt(\vec{u}) \mid \vec{u} \in \mathcal{C}, \vec{u} \neq 0\right\}.\) The minimum distance \(d\) is equal to the minimum non-zero Hamming weight for linear codes. Let \(\mathcal{C}^{\perp_{h}}=\{\vec{v} \in \mathbb{F}_{q^{2}}^{n} \mid\langle\vec{u}, \vec{v}\rangle_{h}=0, \forall \vec{u} \in \mathcal{C}\}\) be the Hermitian dual code of \(\mathcal{C}\). If \(\mathcal{C} \subseteq \mathcal{C}^{\perp_{h}}\), then we say that \(\mathcal{C}\) is a Hermitian self-orthogonal code and that \(\mathcal{C}^{\perp_{h}}\) is a Hermitian dual-containing code.

For a matrix \(B \in \mathbb{F}_{q^2}^{m \times r}\) with full column rank, a matrix \(D \in \mathbb{F}_{q^2}^{m \times r}\) is called a left inverse of \(B\) if \(D^{T}B = I_r\), where \(I_r\) denotes the \(r \times r\) identity matrix. The columns of \(D\) form a biorthogonal basis to the columns of \(B\), i.e., \(d_i^{T}b_j = \delta_{ij}\) for all \(1 \leq i,j \leq r\), where \(\delta_{ij}\) is the Kronecker delta.

2.1 Quasi-cyclic code↩︎

We define the quotient ring \(\mathcal{R}=F_{q^{2}}[x] /\left\langle x^{n}-1\right\rangle\). If \(\mathcal{C}\) is generated by a monic divisor \(g(x)\) of \(x^{n}-1\), i.e., \(\mathcal{C}=\langle g(x)\rangle\) and \(g(x)\mid x^n-1\), then \(g(x)\) is called generator polynomial of \(\mathcal{C}\). For any \(c=\left(c_{0}, c_{1}, \ldots, c_{n-1}\right)\in \mathcal{C}\), we can say \(\mathcal{C}\) is a cyclic code if \(c^\prime =\left(c_{n-1}, c_{0}, \ldots, c_{n-2}\right)\in \mathcal{C}\).

Let \(\Omega_{n}=\{0,1,\ldots,n-1\}\), and let \(\gamma\) be a primitive \(n\)-th root of unity in some extension field of \(\mathbb{F}_{q^{2}}\), where \(n\) is odd. The defining set \(T\) of \(\mathcal{C}=\langle g(x)\rangle\) is denoted as \(T=\left\{i \in \Omega_{n} \mid g\left(\gamma ^{i}\right)=0\right\}\).

Let \(i\) be an integer with \(0\leq i < n\), the set \(C_{i}=\{i, q^{2}i, (q^{2})^{2}i, \cdots , (q^{2})^{k-1}i \}\) (mod \(n\)) is called the \(q^{2}\)-cyclotomic coset modulo \(n\) that contains \(i\), where \(k\) is the smallest positive integer such that \((q^{2})^{k}i\) \(\equiv i\) (mod \(n\)). For each \(i \in \Omega_{n}\), a cyclotomic coset \(C_{i}\) is skew symmetric if \(n-qi\)(mod \(n\)) \(\in C_{i}\), and is skew asymmetric otherwise. Skew asymmetric cosets \(C_{i}\) and \(C_{n-qi}\) come in pair, we use \((C_{i},C_{n-qi})\) to denote such a pair. Let \(\mathcal{C}\) be a cyclic code with a defining set \(T = \bigcup_{i \in \Omega_{n}} C_{i}\). Denoting \(T^{-q}=\{n-qx | x\in T \}\), then we can deduce that the defining set of \(\mathcal{C}\) \(^{\bot _{h}}\) is \(T^{\perp _{h}} =\) \(\mathbb{Z}_{n}\) \(\backslash T^{-q}\). If \(T \cap T^{-q}=\emptyset\), then \(\mathcal{C}^{\perp_{h}} \subseteq \mathcal{C}\), i.e., \(g(x)\mid g^{\perp_h}(x)\).

\(\mathcal{C}\) is said to be quasi-cyclic of index \(l\) or \(l\)-quasi-cyclic if a cyclic shift of any codeword by \(l\) positions is also a codeword in \(\mathcal{C}\). The length \(n\) of a quasi-cyclic code \(\mathcal{C}\) is a multiple of \(l\), i.e., \(n=ml\). The generator matrix of quasi-cyclic code is composed of circulant matrices. It means that a 1-generator quasi-cyclic code can be transformed into an equivalent code with a generator matrix \[G = (A_{0}, A_{1}, A_{2}, \ldots, A_{l})\] where \(A_{i}, i = 0,1,\ldots,l\) is defined as \(m\times m\) circulant matrix \[A=\left(\begin{array}{ccccc} a_{0} & a_{1} & a_{2} & \ldots & a_{m-1} \\ a_{m-1} & a_{0} & a_{1} & \ldots & a_{m-2} \\ a_{m-2} & a_{m-1} & a_{0} & \ldots & a_{m-3} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ a_{1} & a_{2} & a_{3} & \ldots & a_{0} \end{array}\right).\]

With a suitable permutation of coordinates, the generator matrix of a 2-generator quasi-cyclic code with index \(l\) can be transformed into the following form. \[G=\left(\begin{array}{ccccccc} A_{1,1} & A_{1,2}& A_{1,3}& \cdots & A_{1,l} \\ A_{2,1} & A_{2,2}& A_{2,3}& \cdots & A_{2,l}\\ \end{array}\right),\] where \(A_{i, j}\) is circulant matrices determined by polynomial \(a_{i, j}(x)\), where \(1 \leq i \leq 2\) and \(1 \leq j \leq l\).

Let \(g(x)=g_{0}+g_{1} x+g_{2} x+\cdots+g_{n-1} x^{n-1} \in \mathcal{R}\) and \([g(x)]=[g_{0},g_{1},g_{2},\cdots,g_{n-1}]\) represents vectors in \(F_{4}^{n}\) determined by the coefficient of \(g(x)\) in an ascending order.

Let \(g(x),h(x),\nu(x)\) be monic polynomials in \(F_{4}[x]\) whose degree is less than \(n\) and such that both \(g(x),h(x)\) divide \(x^{n}-1\). Any cyclic codes of length \(n\) and dimension \(n-deg(g)\) can be generated by \(\langle g(x)\rangle\) . We consider the check polynomial \(h(x)\) such that \(g(x)\cdot h(x)=x^{n}-1\). Attached to a polymial \(r(x)=r_{0}+r_{1}x+\dots+r_{m}x^{m}\) of degree \(m<n\), we can define \(r^{[2]}(x)=r_{0}^{2}+r_{1}^{2}x+\dots+r_{m}^{2}x^{m}\).

It is well-known that \(\langle g^{\perp _{h}}(x)\rangle\) generates the Hermitian dual code of cyclic code which is generated by \(\langle g(x)\rangle\).

2.2 Quantum codes↩︎

A \(q\)-ary quantum error-correcting codes \(\mathcal{Q}\) of length \(n\) is a \(K\)-dimensional subspace of \(q^n\)-dimensional Hilbert space \((\mathbb{C}^{q})^{\otimes n}\), where \(\mathbb{C}\) represents complex field and \((\mathbb{C}^{2})^{\otimes n}\) is the \(n\)-fold tensor power of \(\mathbb{C}^q\). A binary quantum codes \(\mathcal{Q}\) can be denoted as \([[n,k,d]]_{q}\), where \(k=log_{q}K\).

Hermitian construction is one of the famous construction methods of quantum code, which established a relationship between quantum codes and classical self-orthogonal codes under the Hermitian inner product. Quantum codes can be derived from other quantum codes by the following propagation rules.

Quantum codes can be derived by the following propagation rules. It will be useful later.

Notation. For \(s\geq 1\) and \(1\leq i\leq s\), denote by \(\mathbf{e}_i\) the \(i\)-th standard basis vector \((0,\ldots,0,1,0,\ldots,0)^{T}\) of \(\mathbb{F}_{q^2}^{s}\). The rank of a matrix \(M\) is denoted by \(\mathop{\mathrm{Rank}}(M)\).

3 General Extending Constructions↩︎

The well-known construction of quantum codes requires classical linear codes to be self-orthogonal or dual-containing. In this section, we investigate a new method for constructing self-orthogonal codes by extending \(r\)-nearly self-orthogonal codes.

Consider the linear code \(\mathcal{C}\), which is an \([n, k]_{q^2}\) code with parity-check matrix \(H\). Define the matrix \(A\) as \(HH^{\dagger}\), where \(A\) is an \((n-k) \times (n-k)\) Hermitian matrix, as shown by the fact that \(A^{\dagger} = A\). This connection serves as an important resource for our analysis. For more details, refer to Section 0.4.6 of [27].

The following lemma directly follows from the previous Wedderburn rank-one reduction formula.

In the lemma, the choice of vector \(\mathbf{u}\) is constrained, making it difficult to find a vector extension that maintains a large dual distance for the constructed code. We introduce the theory of matrix spectral decomposition and employ a method that utilizes the matrix’s eigen space to find the vector \(\mathbf{u}\).

The general rank-reduction principle above can be instantiated concretely using the Jordan canonical decomposition of \(A\). Let \(A = P J_A P^{-1}\) be the Jordan canonical form of \(A\), and set \(W = P^{-1}\). For Hermitian matrices over \(\mathbb{F}_{q^2}\), the Jordan structure is strictly constrained: all nonzero eigenvalues lie in \(\mathbb{F}_q^*\) and correspond to 1-by-1 Jordan blocks, while all nilpotent Jordan blocks have size at most 2. Each Jordan block (whether a semisimple 1-block or a 2-step nilpotent block) contributes exactly one to the rank of \(A\).

To perform the \(i\)-th rank reduction we select a column \(\boldsymbol{w}_j\) of \(W = P^{-1}\) and verify the active condition \[\label{eq:active} A \boldsymbol{w}_j \neq \mathbf{0}.\tag{1}\] Condition 1 is equivalent to \(\mathop{\mathrm{Rank}}(A + \boldsymbol{w}_j\boldsymbol{w}_j^\dagger) = \mathop{\mathrm{Rank}}(A) - 1\), so it guarantees a valid rank-1 reduction by Theorem 3. When \(J_A\) contains a 1-by-1 block with nonzero eigenvalue \(\lambda \in \mathbb{F}_q^*\), the corresponding column \(\boldsymbol{w}_j\) of \(W\) satisfies \(A\boldsymbol{w}_j = \lambda P \mathbf{e}_j \neq \mathbf{0}\); hence such columns always satisfy 1 . For a size-2 nilpotent block the active condition still selects a suitable column, but see Remark [rem:nilpotent] for the special algebraic structure in that case.

In our computational implementation the candidate columns of \(W\) are examined in increasing Hamming-weight order; the first column satisfying 1 and preserving the dual distance \(d\) is appended to the current parity-check matrix. This weight-first heuristic is a search optimization and is not required by the theory.

Now, using the previous iterative rank reduction method, we can present the generation method for self-orthogonal codes from arbitrary codes.

Next, we determine a dual-distance preservation criterion for these extended Hermitian self-orthogonal codes.

Let \(\mathcal{C}=[n,k,d]_{q^2}\) be a classical linear code over \(\mathbb{F}_{q^2}\) with parity-check matrix \(H\in\mathbb{F}_{q^2}^{(n-k)\times n}\). Define the Hermitian matrix \[A = HH^{\dagger} \in \mathbb{F}_{q^2}^{(n-k)\times (n-k)},\] where \(H^{\dagger}\) denotes the Hermitian conjugate transpose. Let \(A=PJP^{-1}\) be the Jordan decomposition with \(W=P^{-1}\).

In Theorem 5 the Jordan basis matrix \(W=P^{-1}\) is the source of all candidate extension vectors. Its columns are computed from the Jordan decomposition \(A=HH^{\dagger}=PJ_AP^{-1}\) of the current Hermitian matrix. Only columns satisfying the active condition (C1) can reduce the rank of \(A\) by one; kernel columns (those with \(A\mathbf{w}_j=\mathbf{0}\)) leave the rank unchanged and must be discarded. In practice we scan the columns of \(W\) in increasing Hamming-weight order and append the first column that satisfies both (C1) and (C2\('\)). This weight-first ordering is a search heuristic and is not required by the theory.

Condition (C1) alone is not sufficient. Consider the BKLC in magma \(\mathcal{C}=[65,44,12]_4\) with \(Rank(A)=1\). For the unique \(\mathbf{w}_j\in \mathop{\mathrm{Im}}(A)\setminus\mathop{\mathrm{Ker}}(A)\), we have \(A\mathbf{w}_j\neq\mathbf{0}\) yet \(d(\mathcal{C}_{\mathrm{ext}}^{\perp_{h}})=10<12\). Here \(supp(\mathbf{w}_j)\) has small overlap with the defining set of \(\mathcal{C}_0\), causing \(NullSpace(H_S^{(q)})\) to contain vectors of weight \(<11\), violating (C2\('\)). This demonstrates that (C2\('\)) is the critical safeguard against distance collapse.

The rank-one Hermitian matrix \(A\) can appear in two Jordan forms:

• Semisimple case: \(J_A=diag(0,\ldots,0,\lambda)\) with \(\lambda\in\mathbb{F}_q^{*}\). Then \(A^2\neq 0\), and we have the direct-sum decomposition \[\mathbb{F}_{q^2}^{n-k}=\mathop{\mathrm{Im}}(A)\oplus\mathop{\mathrm{Ker}}(A).\] Condition (C1) is equivalent to \(\mathbf{w}_j\in\mathop{\mathrm{Im}}(A)\setminus\{0\}\). Among the columns of \(W=P^{-1}\), exactly those spanning \(\mathop{\mathrm{Im}}(A)\) satisfy (C1); every kernel column is mapped to zero.

• Nilpotent case: \(J_A\) is a single \(2\times 2\) nilpotent Jordan block (with all other diagonal entries zero). Then \(A^2=0\), which implies \(\mathop{\mathrm{Im}}(A)\subseteq\mathop{\mathrm{Ker}}(A)\). Consequently the set difference \(\mathop{\mathrm{Im}}(A)\setminus\mathop{\mathrm{Ker}}(A)\) is empty, and any characterization of condition (C1) as “\(\mathbf{w}_j\in\mathop{\mathrm{Im}}(A)\setminus\mathop{\mathrm{Ker}}(A)\)” becomes vacuous. This is precisely why Theorem 5 assumes a non-zero eigenvalue.

In the nilpotent case one must therefore rely on the Jordan-basis formulation: select a column \(\mathbf{w}_j\) of \(W=P^{-1}\) and verify the active condition \(A\mathbf{w}_j\neq\mathbf{0}\) directly. Only such columns guarantee the rank drop \(\mathop{\mathrm{Rank}}(A+\mathbf{w}_j\mathbf{w}_j^{\dagger})=\mathop{\mathrm{Rank}}(A)-1\). Our implementation always performs this direct rank check, so it handles both semisimple and nilpotent Jordan blocks uniformly.

4 New Quantum Codes↩︎

4.1 Jordan Extension of Dual-Containing Codes from Cyclic Codes↩︎

Example 1 \([[53,25,8]]_{2}\) quantum code. Let \(n = 51\), \(q = 2\). Factor \(x^{51}-1\) over \(\mathbb{F}_4 = \mathbb{F}_2[\omega]/\langle \omega^2 + \omega + 1 \rangle\). The 4-cyclotomic cosets modulo 51 include \[\begin{align} C_{0} &= \{0\}, \\ \quad C_{1} = \{1, 4, 16, 13\},\\ \quad C_{2} = \{2, 8, 32,26\}, \\ \quad C_{3} = \{3, 12, 48,39\}, \\ \quad C_{5} = \{5, 20, 29,14\}, \\ \quad\cdots\\ \quad C_{34} = \{34\}. \end{align}\] Take the generator polynomial \[\begin{gather} g(x)=M_1(x)M_3(x)M_5(x)M_{34}(x)M_0(x)\\ =x^{14} + \omega^2 x^{13} + \omega x^{11} + x^{10} + \omega x^{9} + x^{8} + \omega^2 x^{7} + \omega x^{6} + \\x^{5} + \omega x^{4} + \omega x^{3} + x^{2} + \omega^2, \end{gather}\] where \(\omega^2+\omega+1=0\). The defining set is \[T = C_1\cup C_3\cup C_5\cup C_{34}\cup C_0,\] and \(\deg g(x)=14\), so \(\mathcal{C}=\langle g(x)\rangle\) is a \([51,37,8]_4\) cyclic code. Its Hermitian dual is \(\mathcal{C}^{\perp}=[51,14,22]_4\); equivalently, a parity-check matrix \(H\) of \(\mathcal{C}\) is a generator matrix of \(\mathcal{C}^{\perp}\), and it is this \(14\times 51\) matrix that we extend.

Step 1: Hermitian deficiency. For a parity-check matrix \(H\) of \(\mathcal{C}\) (equivalently a generator matrix of \(\mathcal{C}^{\perp}\)), form \(A=HH^{\dagger}\in\mathbb{F}_4^{14\times 14}\). Direct computation gives \[r=\mathop{\mathrm{Rank}}(A)=\mathop{\mathrm{Rank}}(HH^{\dagger})=2.\] Thus two extension columns are needed to obtain a Hermitian self-orthogonal code.

Step 2: Jordan decomposition. For each extension step we decompose the relevant matrix \(A_i\) as \(A_i = P_i J_i P_i^{-1}\) and set \(W_i = P_i^{-1}\) (\(i=1,2\)). The Jordan forms are \[J_1 = \operatorname{diag}\bigl(\underbrace{0,\ldots,0}_{11\text{ zeros}},\, J_2(0),\, 1\bigr), \qquad J_2(0) = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\] for the first extension step, and \[J_2 = \operatorname{diag}\bigl(\underbrace{0,\ldots,0}_{13\text{ zeros}},\, 1\bigr),\] for the second extension step. In both cases the only nonzero eigenvalue is \(1 \in \mathbb{F}_2^{*} = \{1\}\). The columns of \(W_i = P_i^{-1}\) form the Jordan basis from which the extension vectors are selected.

Step 3: Iterative extension preserving the dual distance. By Theorem 6, at each step we recompute the Jordan decomposition \(A_i=P_iJ_{A_i}P_i^{-1}\) of the current Hermitian matrix, form \(W_i=P_i^{-1}\), and choose a column \(\mathbf{w}_j\) of \(W_i\) satisfying (C1) \(A_i\mathbf{w}_j\neq\mathbf{0}\) and (C2\('\)). We append it to \(H\) and verify that the rank drops by one while the dual distance stays equal to \(8\).

• Extension 1: Select column \(12\) of \(W_1=P_1^{-1}\), which has Hamming weight \(12\): \[\mathbf{w}^{(1)}=(\omega^2,\omega^2,\omega,0,\omega,\omega^2,1,\omega,\omega^2,1,0,\omega,\omega,1)^{T}.\] Appending it gives \(H_1=[H\mid\mathbf{w}^{(1)}]\). Direct computation yields \[\mathop{\mathrm{Rank}}(H_1H_1^{\dagger})=2\to 1, \qquad d\bigl(\langle H_1\rangle^{\perp_h}\bigr)=8.\] Thus the first extension satisfies both (C1) and (C2\('\)) for \(d=8\).

• Extension 2: Recompute \(W_2=P_2^{-1}\) from \(A_2=H_1H_1^{\dagger}\) and select column \(14\), which has Hamming weight \(9\): \[\mathbf{w}^{(2)}=(1,\omega,1,1,1,0,\omega^2,0,\omega,0,\omega,0,0,1)^{T}.\] Appending it gives \(H_2=[H_1\mid\mathbf{w}^{(2)}]\). Direct computation yields \[\mathop{\mathrm{Rank}}(H_2H_2^{\dagger})=1\to 0, \qquad d\bigl(\langle H_2\rangle^{\perp_h}\bigr)=8.\] Hence the second extension also preserves the dual distance.

Step 4: Quantum code parameters. The resulting code is Hermitian self-orthogonal with parameters \(\mathcal{C}_{\mathrm{so}}=[53,14,24]_4\); its Hermitian dual has parameters \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[53,39,8]_4\) with weight enumerator polynomial \(W_{39,53}(z)=\) \(1+29682z^{8}+290088z^{9}+4440672z^{10}+50044464z^{11}+526129260z^{12}+\dots+72208218802902408z^{53}\). Applying the Hermitian construction yields \[\mathcal{Q}=[[53,\, 2\cdot 37 - 51 + 2,\, 8]]_2 = \mathbf{[[53, 25, 8]]_2}.\] This improves the previous best-known quantum code \([[53,25,7]]_2\) listed in Grassl’s tables [31] to distance \(8\).

By the propagation rules in Proposition [propagation95rule], \([[53, 25, 8]]_2\) also yields the additional quantum code \([[54,25,8]]_2\) (by lengthening).

Table 1 lists further record quantum codes obtained by the same Jordan-extension procedure from cyclic, quasi-cyclic, and BCH codes over \(\mathbb{F}_{q^2}\). In each case the extension column(s) are selected from \(W=P^{-1}\) of the current Hermitian matrix \(A=HH^{\dagger}\); the active condition (C1) and the nullspace condition (C2\('\)) are verified at every step, so the dual distance is preserved.

In this section we describe the six record codes of Table 1 individually. Each example follows the same Jordan-extension pipeline as the \([[53,25,8]]_2\) code above: compute \(A=HH^{\dagger}\), form its Jordan decomposition \(A=PJ_AP^{-1}\), select suitable columns of \(W=P^{-1}\) satisfying (C1) and (C2\('\)), append them to \(H\), and verify that the rank drops by one and the dual distance is preserved at every step.

Example 2 \([[59,29,8]]_2\). Consider the following polynomials in \(R\): \[f(x) = x^{14} + \omega x^{13} + \omega x^{11} + \omega^2 x^{10} + x^9 + \omega^2 x^8 + \omega x^7 + \omega^2 x^6 +x^5\] \[+ \omega^2 x^4 + \omega x^3 + \omega x + 1,\] \[v_2(x) = \omega^2 x^{26} + x^{25} + x^{23} + \omega^2 x^{22} + \omega x^{21} + \omega x^{20} + \omega x^{18} + \omega x^{16} + x^{15}+\] \[\omega x^{14} + \omega x^{13} + \omega^2 x^{12} + \omega^2 x^{10} + \omega x^8 + \omega^2 x^6 + \omega^2 x^5 + x^3+ \omega^2 x^2 + x + \omega,\]

and \[v_2(x) = \omega x^{28} + \omega^2 x^{27} + x^{24} + x^{23} + x^{21} + \omega x^{20} + x^{19} + \omega x^{18} + \omega^2 x^{17} +\] \[\omega x^{16} + \omega^2 x^{15} + \omega^2 x^{14} + \omega^2 x^{12} + x^{10} + \omega^2 x^8 + \omega^2 x^6 + \omega^2 x^5 + x^4 + \omega x^3 + \omega.\]

We define a 1-quasi-cyclic code of length \(2n\) \(\mathcal{C}=[58,43,8]_4\) code \[\mathcal{C} = \big\langle (v_1 f,\; v_2 f_1)\big.\rangle\] Its parity-check matrix \(H\) has size \(15\times 58\), and \[A=HH^{\dagger}=\begin{pmatrix} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \end{pmatrix}_{15\times 15}, r=Rank(A)=1.\] Compute \(A=PJ_AP^{-1}\) and put \(W=P^{-1}\). The selected column of \(W\) satisfying (C1) \(A\mathbf{w}^{(1)}\neq\mathbf{0}\) and (C2\('\)) \(d(NullSpace(H_S^{(2)}))\ge 7\) with \(S=supp(\mathbf{w}^{(1)})\) is \[\mathbf{w}=(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)^{T}\in\mathbb{F}_4^{15}.\] Appending it to \(H\) gives \(H_1=[H\mid\mathbf{w}^{(1)}]\) with \[\mathop{\mathrm{Rank}}(H_1H_1^{\dagger})=1\to 0,\qquad d\bigl(\langle H_1\rangle^{\perp_h}\bigr)=8.\] Thus we obtain a self-orthogonal code \(\mathcal{C}_{\mathrm{so}}=[59,15,26]_4\) whose Hermitian dual is \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[59,44,8]_4\)with weight enumerator polynomial \(W_{44,59}(z)=\) \(1+13746z^{8}+229767z^{9}+3451638z^{10}+46197174z^{11}+554418984z^{12}+6008492136z^{13}+\dots+13159947574445658066z^{59}\). Hence \[\mathcal{Q}=[[59,\,2\cdot 43-58+1,\,8]]_2=\mathbf{[[59,29,8]]_2},\] improving the previously known parameters in Grassl’s tables. By the propagation rules in Proposition [propagation95rule], \([[59,29,8]]_2\) also yields the additional quantum codes \([[60,29,8]]_2\) (by lengthening) and \([[61,29,8]]_2\) (by subcode construction).

Example 3 \([[79,41,9]]_2\). Let \(\mathcal{C}\) be the \((2,2)\)-quasi-cyclic \([76,57,9]_4\) code with generator polynomials \(f_{1}(x) = x + \omega^{2}\), \(f_{2}(x) = x^{18} + \omega x^{17} + \omega x^{16} + x^{14} + x^{13} + \omega^{2}x^{11} + \omega^{2}x^{9} + x^{7} + \omega^{2}x^{5} + \omega x^{4} + x^{2} + \omega^{2} x + 1\), and a fixed \(v(x) = x^{38} +\omega x^{37} +\omega x^{34} + \omega^2 x^{32} + \omega^2 x^{31} + x^{29} + \omega x^{28} + \omega^{2}x^{26} + \omega x^{25} + \omega x^{24} + \omega^{2} x^{23} + \omega x^{22} + x^{21} + \omega^{2} x^{20} + \omega^{2} x^{19} + \omega^{2} x^{18} + \omega^{2} x^{17} + \omega x^{16} + \omega x^{13} + x^{12} + x^{5} + x^{4} + \omega x^{3} + x + 1\).

Thus \(\mathcal{C}\) is the 2-quasi-cyclic code of length \(2n=76\) generated as an \(R\)-submodule by \((v f_1,\; f_1)\) and \((f_2,\; v f_2)\), or equivalently by the \(2\times 2\) polynomial generator matrix \[G(x) = \begin{pmatrix} v f_1 & f_1 \\[2mm] f_2 & v f_2 \end{pmatrix} \in \mathcal{M}_{2 \times 2}(R).\] Its Hermitian dual \(\mathcal{C}^{\perp_h}\) has dimension \(19\); taking a parity-check matrix \(H\) of \(\mathcal{C}\) (equivalently a generator matrix of \(\mathcal{C}^{\perp_h}\)) we compute \(A=HH^{\dagger}\in\mathbb{F}_4^{19\times 19},\qquad r=\mathop{\mathrm{Rank}}(A)=3\).

Compute \(A=PJ_AP^{-1}\) and put \(W=P^{-1}\). The three extension columns are selected iteratively from the respective Jordan bases; one of the selected columns satisfying (C1) \(A\mathbf{w}^{(1)}\neq\mathbf{0}\) and (C2\('\)) \(d(NullSpace(H_S^{(2)}))\ge 9\) with \(S=supp(\mathbf{w}^{(1)})\) is \[\mathbf{w}=(1,1,\omega,\omega,0,1,0,\omega^{2},1,0,\omega^{2},1,1,0 ,\omega,\omega^{2},1,1,\omega)^{T}.\] After the three iterative extensions the rank of the Hermitian product matrix is reduced from \(3\) to \(0\), and the extended dual code has minimum distance \(9\). The resulting self-orthogonal code has parameters \(\mathcal{C}_{\mathrm{so}}=[79,19,\cdot]_4\) and Hermitian dual \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[79,60,9]_4\) with weight enumerator polynomial \(W_{60,79}(z)=\) \(1+12909z^{9}+309231z^{10}+5804838z^{11}+99095841z^{12}+1530748683z^{13}+\dots+179241796303546028802673698z^{79}\). Therefore \[\mathcal{Q}=[[79,\,2\cdot 57-76+3,\,9]]_2=\mathbf{[[79,41,9]]_2},\] improving the previously known parameters in Grassl’s tables. By the propagation rules in Proposition [propagation95rule], \([[79,41,9]]_2\) also yields the additional quantum code \([[80,41,9]]_2\) (by lengthening).

Example 4 \([[92,78,5]]_3\). Let \(\mathcal{C}\) be the BCH code \([91,84,5]_9\) with defining set constructed in the usual way over \(\mathbb{F}_9=\mathbb{F}_3[\omega]\). Its parity-check matrix \(H\) has size \(7\times 91\), and \[A=HH^{\dagger}\in\mathbb{F}_9^{7\times 7},\qquad r=\mathop{\mathrm{Rank}}(A)=1.\] The Jordan form has one non-zero eigenvalue in \(\mathbb{F}_3^{*}\). Forming \(W=P^{-1}\), the corresponding column satisfying (C1) and (C2\('\)) for \(d=5\) is \[\mathbf{w}^{(1)}=(1,\omega^5,0,2,\omega,\omega^6,1)^{T}\in\mathbb{F}_9^{7}.\] Appending it to \(H\) gives \(H_1=[H\mid\mathbf{w}^{(1)}]\) with \[\mathop{\mathrm{Rank}}(H_1H_1^{\dagger})=1\to 0,\qquad d\bigl(\langle H_1\rangle^{\perp_h}\bigr)=5.\] The extended self-orthogonal code has parameters \(\mathcal{C}_{\mathrm{so}}=[92,7,70]_9\); its Hermitian dual is \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[92,85,5]_9\) with weight enumerator polynomial \(W_{85,92}(z)=1+74970z^{5}+5542680z^{6}+\dots\). Consequently \[\mathcal{Q}=[[92,\,2\cdot 84-91+1,\,5]]_3=\mathbf{[[92,78,5]]_3},\] improving the previously known parameters in Grassl’s tables. By the propagation rules in Proposition [propagation95rule], \([[92,78,5]]_3\) also yields the additional quantum codes \([[93,78,5]]_3\) (by lengthening).

Example 5 \([[80,64,6]]_3\). Let \(\mathcal{C}=[79,71,6]_9\) be the best-known linear code described in Grassl’s tables. We have \[A = H H^{\dagger} \in \mathbb{F}_9^{8\times 8}, \qquad r = \operatorname{Rank}(A) = 1,\] where \[A = \begin{pmatrix} 2 & 0 & \omega^7 & \omega^7 & \omega^6 & \omega^2 & \omega^5 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \omega^5 & 0 & 1 & 1 & \omega^7 & \omega^3 & \omega^6 & \omega \\ \omega^5 & 0 & 1 & 1 & \omega^7 & \omega^3 & \omega^6 & \omega \\ \omega^2 & 0 & \omega^5 & \omega^5 & 2 & 1 & \omega^3 & \omega^6 \\ \omega^6 & 0 & \omega & \omega & 1 & 2 & \omega^7 & \omega^2 \\ \omega^7 & 0 & \omega^2 & \omega^2 & \omega & \omega^5 & 1 & \omega^3 \\ 1 & 0 & \omega^3 & \omega^3 & \omega^2 & \omega^6 & \omega & 2 \end{pmatrix}.\] with \(H\) of size \(8\times 79\). Computing \(A=PJ_AP^{-1}\) and \(W=P^{-1}\), the selected column satisfying (C1) and (C2\('\)) for \(d=6\) is \[\mathbf{w}=(1,0,\omega,\omega,\omega^6,\omega^2,\omega^3,2)^{T}\in\mathbb{F}_9^{8}.\] Appending it to \(H\) gives \(H_1=[H\mid\mathbf{w}^{(1)}]\) with \[\mathop{\mathrm{Rank}}(H_1H_1^{\dagger})=1\to 0,\qquad d\bigl(\langle H_1\rangle^{\perp_h}\bigr)=6,\] and we obtain a self-orthogonal code \(\mathcal{C}_{\mathrm{so}}=[80,8,56]_9\) whose Hermitian dual has parameters \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[80,72,6]_9\) with weight enumerator polynomial \(W_{72,80}(z)=\) \(1+1929192z^{6}+139704824z^{7}+10180184456z^{8}+641595165672z^{9}+35936588753792 z^{10}+\dots\). Thus \[\mathcal{Q}=[[80,\,2\cdot 71-79+1,\,6]]_3=\mathbf{[[80,64,6]]_3},\] improving the previously known parameters in Grassl’s tables. By the propagation rules in Proposition [propagation95rule], \([[80,64,6]]_3\) also yields the additional quantum codes \([[81,64,6]]_3\) (by lengthening) and \([[80,63,6]]_3\) (by subcode construction).

Example 6 \([[131,93,8]]_2\). Let \(\mathcal{C}\) be the degree-2 quasi-cyclic \([130,111,8]_4\) code with generator polynomials \(f_1(x) = x^{46} + \omega^{2} x^{44} +\omega^{2} x^{43} + \omega x^{40} + x^{38} +\omega x^{37} +\omega x^{36} + x^{35} + x^{34} + \omega x^{33} + \omega x^{32} + x^{28} + \omega x^{25} + \omega x^{23} + \omega x^{21} + x^{18} + \omega x^{14} +\omega x^{13} + x^{12} + x^{11} + \omega x^{10} + \omega x^{9} + x^{8} + \omega x^{6} + \omega^{2} x^{3} + \omega^{2} x^{2} + 1\),

\(f_2(x) = \omega x^{18} + x^{16} + x^{13} + x^{12} + \omega^{2} x^{9} + \omega x^{7} + \omega x^{6} + \omega^{2} x^{5} + \omega x^{3} + x\),

\(f_{3}(x) = x^{65} + 1\), and a fixed \(v(x) = x^{46} + \omega^2 x^{44} + \omega^2 x^{43} + \omega x^{40} + x^{38} +\omega x^{37} + \omega x^{36} + x^{35} + x^{34} + \omega x^{33} + \omega x^{32} + x^{28} + \omega x^{25} + \omega x^{23} + \omega x^{21} + x^{18} +\omega x^{14} + \omega x^{13} + x^{12} + x^{11} +\omega x^{10} + \omega x^{9} + x^{8} + \omega x^{6} + \omega^2 x^{3} + \omega^2 x^{2} + 1\).

Therefore, a 2-quasi-cyclic code of length \(2n\) is an \(R\)-submodule of \(R^2\). Let \(n=65\). We define \[\mathcal{C} = \big\langle ( f_1,\; v f_2),\; (0,\; f_3) \big\rangle_R \subseteq R^2, \label{eq:C95R95submodule95ex7}\tag{5}\] Equivalently, \(\mathcal{C}\) is the \(2 \times 2\) polynomial generator matrix \[G(x) = \begin{pmatrix} f_1 & f_2 \\[2mm] 0 & f_3 \end{pmatrix} \in \mathcal{M}_{2 \times 2}(R). \label{eq:poly95G95ex7}\tag{6}\] Its Hermitian dual \(\mathcal{C}^{\perp_h}\) has dimension \(19\); taking a parity-check matrix \(H\) of \(\mathcal{C}\) (equivalently a generator matrix of \(\mathcal{C}^{\perp_h}\)) we compute \(A=HH^{\dagger}\in\mathbb{F}_4^{19\times 19},\qquad r=\mathop{\mathrm{Rank}}(A)=1\). The Jordan form has one non-zero eigenvalue in \(\mathbb{F}_2^{*}\). Forming \(W=P^{-1}\), the selected column satisfying (C1) and (C2\('\)) for \(d=8\) is \[\mathbf{w}=(1,1,\omega,1,\omega^2,\omega^2,\omega,\omega^2,0,\omega^2,0,\omega^2, \omega,\omega^2,\omega^2,1,\omega,1,1)^{T}\in\mathbb{F}_4^{19}.\] Appending it to \(H\) gives \(H_1=[H\mid\mathbf{w}]\) with \[\mathop{\mathrm{Rank}}(H_1H_1^{\dagger})=1\to 0,\qquad d\bigl(\langle H_1\rangle^{\perp_h}\bigr)=8.\] The extended self-orthogonal code has parameters \(\mathcal{C}_{\mathrm{so}}=[131,19,66]_4\); its Hermitian dual is \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[131,112,8]_4\) with weight enumerator polynomial \(W_{112,131}(z)=\) \(1+38610z^{8}+1700400z^{9}+61806030z^{10}+2044336710z^{11}+61342419840 z^{12}+1684397370060 z^{13}+42592353523800 z^{14}+\dots\). Consequently \[\mathcal{Q}=[[131,\,2\cdot 111-130+1,\,8]]_2=\mathbf{[[131,93,8]]_2},\] improving the previously known parameters in Grassl’s tables.

By the propagation rules in Proposition [propagation95rule], Example 7 also yields the additional quantum codes \([[132,93,8]]_2\), \([[133,93,8]]_2\), \([[134,93,8]]_2\), \([[135,93,8]]_2\) (by lengthening) and \([[131,92,8]]_2\), \([[132,92,8]]_2\), \([[133,92,8]]_2\) (by subcode construction).

Example 7 \([[171,149,5]]_2\). Let \(\mathcal{C}\) be the \([170,159,5]_4\) quasi-cyclic code with generator polynomials \(f_{1}(x) = x^{10} + \omega^{2} x^{9} + \omega x^{8} + \omega x^{7} + x^{6} + x^{2} + x + 1\), \(f_{2}(x) = x + 1\), and a fixed \(v(x) = \omega x^{84} + \omega^2 x^{80} + \omega x^{79} + \omega^2 x^{78} + x^{77} + \omega^2 x^{76} + x^{75} + \omega x^{74} \\ + \omega^2 x^{73} + \omega x^{72} + \omega^2 x^{71} + x^{70} + x^{69} + x^{66} + \omega x^{65} + x^{64} + \omega^2 x^{63} + x^{62} + \omega^2 x^{58} + \omega x^{57} + \omega x^{56} + x^{54} + x^{53} + \omega^2 x^{52} + \omega^2 x^{51} \\ + \omega x^{50} + \omega^2 x^{49} + \omega x^{48} + \omega^2 x^{47} + x^{46} + \omega x^{45} + x^{44} + \omega x^{43} + \omega^2 x^{42} + x^{41} + \omega^2 x^{40} + x^{39} + \omega x^{38} + x^{37} + \omega^2 x^{36} + \omega^2 x^{34} \\ + \omega x^{32} + \omega^2 x^{31} + x^{30} + \omega^2 x^{29} + \omega x^{28} + \omega x^{27} + \omega x^{25} + x^{23} + \omega^2 x^{22} + \omega^2 x^{21} + \omega^2 x^{20} + \omega x^{19} + x^{17} + x^{16} + \omega x^{15} + \omega x^{14} \\ + \omega x^{13} + \omega^2 x^{12} + x^{11} + x^{10} + \omega^2 x^9 + \omega^2 x^8 + \omega^2 x^7 + \omega^2 x^5 + \omega x^4 + x^2 + x + 1\).

Therefore, a 2-quasi-cyclic code of length \(2n\) is an \(R\)-submodule of \(R^2\). Let \(n=85\). We define \[\mathcal{C} = \big\langle (v f_1,\; f_1),\; (f_2,\; v f_2) \big\rangle_R \subseteq R^2, \label{eq:C95R95submodule}\tag{7}\] i.e., \(\mathcal{C}\) is generated as an \(R\)-module by the two rows \[\mathbf{g}_1 = (v f_1,\; f_1), \qquad \mathbf{g}_2 = (f_2,\; v f_2).\] Equivalently, \(\mathcal{C}\) is the image of the \(2 \times 2\) polynomial generator matrix \[G(x) = \begin{pmatrix} v f_1 & f_1 \\[2mm] f_2 & v f_2 \end{pmatrix} \in \mathcal{M}_{2 \times 2}(R). \label{eq:poly95G}\tag{8}\] Its Hermitian dual \(\mathcal{C}^{\perp_h}\) has dimension \(11\); taking a parity-check matrix \(H\) of \(\mathcal{C}\) (equivalently a generator matrix of \(\mathcal{C}^{\perp_h}\)) we compute \(A=HH^{\dagger}\in\mathbb{F}_4^{11\times 11},\qquad r=\mathop{\mathrm{Rank}}(A)=1\).

The Jordan form \(J_A\) is nilpotent (all eigenvalues are \(0\)), so this is the nilpotent rank-one case discussed in Remark [rem:nilpotent]. We therefore form \(W=P^{-1}\) and scan its columns; the active condition (C1) must be checked directly because \(\mathop{\mathrm{Im}}(A)\subseteq\mathop{\mathrm{Ker}}(A)\). The selected column of \(W=P^{-1}\) is \(\mathbf{w}=(1,1,1,0,0,0,1,\omega,\omega,\omega^2,1)^{T}\in\mathbb{F}_4^{11}.\) It satisfies \(A\mathbf{w}^{(1)}\neq\mathbf{0}\) and the nullspace condition (C2\('\)) for \(d=5\). Appending it to \(H\) gives \(H_1=[H\mid\mathbf{w}^{(1)}]\) and \(\mathop{\mathrm{Rank}}(H_1H_1^{\dagger})=1\to 0,\qquad d\bigl(\langle H_1\rangle^{\perp_h}\bigr)=5.\) The resulting Hermitian self-orthogonal code has parameters \(\mathcal{C}_{\mathrm{so}}=[171,11,86]_4\); its Hermitian dual is \(\mathcal{C}_{\mathrm{so}}^{\perp_h}=[171,160,5]_4\) with weight enumerator polynomial \(W_{160,171}(z)=\) \(1+74970z^{5}+5542680z^{6}+390228285z^{7}+24029742165z^{8}+1305147006270z^{9}+63429558626310z^{10}+\dots\).

The Hermitian construction therefore yields \[\mathcal{Q}=[[171,\,2\cdot 159-170+1,\,5]]_2=\mathbf{[[171,149,5]]_2}.\] This improves the previously known parameters in Grassl’s tables for \(\mathcal{Q}=\mathbf{[[171,149,4]]_2}\). By the propagation rules in Proposition [propagation95rule], \([[171,149,5]]_2\) also yields the additional quantum codes \([[172,149,5]]_2\) (by lengthening).

To assess the robustness of the construction, we implemented the iterative Jordan-extension procedure in Magma [32] and ran it over a large family of cyclic codes. In every computed instance the dual distance of the extended code was at least the dual distance of the original code; Table 2 lists the six records obtained for length \(n=51\). In particular, the cyclic code \([51,36,8]_4\) with defining set \(T=C_1\cup C_3\cup C_5\cup C_{17}\cup C_{34}\cup C_0\) has Hermitian deficiency \(r=3\) and yields the quantum code \([[54,24,8]]_2\); its three extension steps selected columns 14, 15, 15 of \(W=P^{-1}\) (weights 13, 13, 13), reducing the rank \(3\to 2\to 1\to 0\) while keeping the dual distance equal to 8 at each step.

A direct application of Theorem 7 cannot produce a code with parameters \([[53,24,8]]_2\), because the formula \(k_Q=2k-n+r\) implies \(k_Q\equiv n+r\pmod 2\), whereas \(53\) is odd and \(24\) is even. Hence the record \([[53,24,8]]_2\) is not attainable by the present construction.

All 2193 constructed quantum codes in our search satisfy the lower bound \(d_Q \geq d_C\) predicted by Theorem 7; for the records in Table 2 equality holds. This provides strong empirical support for the sufficiency of the rank-reduction and distance-preservation criteria in Theorem 5.

According to Corollary 1, we obtain new binary quantum codes from the Hermitian dual-containing codes constructed above. By Proposition [propagation95rule], applying lengthening and subcode constructions to these codes produces additional new binary quantum codes, all of which improve the lower bounds on the minimum distance in Grassl’s table [31]. The parameters of the new quantum codes obtained by direct Jordan extension are shown in Table 2, and further codes derived by propagation rules are listed in Table ¿tbl:tab:new95quantum95codes?.

5 Conclusion↩︎

A recurring obstacle in the construction of stabilizer quantum codes from classical linear codes is the Hermitian self-orthogonality requirement \(\mathcal{C}^{\perp_h}\subseteq\mathcal{C}\). In this paper we have introduced a Jordan-canonical-form framework that removes this obstruction for arbitrary \([n,k,d]_{q^2}\) linear codes by explicitly extending the parity-check matrix \(H\) with \(r=\mathop{\mathrm{Rank}}(HH^\dagger)\) additional columns. The decomposition \(A=HH^\dagger=PJ_AP^{-1}\) turns the rank-reduction task into a linear-algebraic problem on the Jordan basis \(W=P^{-1}\): each extension step selects a column of \(W\) that satisfies the active condition \(A\mathbf{w}_j\neq\mathbf{0}\), thereby decreasing the rank of the Hermitian product matrix by exactly one. After \(r\) steps the augmented matrix \(H_r\) generates a Hermitian self-orthogonal code with parameters \([n+r,n-k]_{q^2}\), and the Hermitian construction yields a \(q\)-ary quantum stabilizer code with parameters \([[n+r,2k-n+r,\ge d]]_q\).

Theoretically, the proposed method differs from earlier extension constructions in two important respects. First, the Jordan basis \(W=P^{-1}\) provides an explicit, deterministic supply of candidate extension vectors; it does not rely on special algebraic structures of the underlying classical code, nor does it require ad hoc choices of extension vectors. Second, Theorem 5 gives a verifiable sufficient criterion—the active condition (C1) together with the nullspace condition (C2\('\))—that guarantees the dual distance is preserved at each step. We also clarified the nilpotent case, where the naive characterization \(\mathbf{w}_j\in\mathop{\mathrm{Im}}(A)\setminus\mathop{\mathrm{Ker}}(A)\) collapses because \(\mathop{\mathrm{Im}}(A)\subseteq\mathop{\mathrm{Ker}}(A)\); in this case condition (C1) must be checked directly on the Jordan basis.

The practical value of the framework is demonstrated by the record quantum codes obtained from cyclic, quasi-cyclic, and BCH codes over \(\mathbb{F}_{q^{2}}\), several of which improve the best-known parameters in Grassl’s tables. These examples show that the Jordan extension not only enlarges the set of classical codes eligible for quantum Construction X, but also preserves the minimum distance without relying on case-by-case algebraic analysis.

Several directions remain open. The Jordan decomposition must be recomputed at each extension step, so a more efficient update rule for the Jordan basis as the matrix grows would be valuable. It would also be natural to extend the same rank-reduction principle to other inner products relevant for quantum codes, such as the Euclidean and symplectic inner products, and to investigate whether the distance-preservation criterion can be sharpened into a necessary and sufficient condition. Finally, combining the Jordan extension with propagation rules and with families of classical codes whose Hermitian hull is well understood may lead to further systematic improvements of quantum code tables.

Acknowledgment↩︎

The authors would like to thank Ruihu Li for the suggestions on our manuscript, which improve the manuscript significantly. This work is supported by the National Natural Science Foundation of China under Grant No.U21A20428, Natural Science Foundation of Shaanxi under Grant No.2025-JC-YBQN-070.

References↩︎

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1. L. Lu,R. Yang, Y. Liu, Q.Fu and G.Guo was with the Department of Basic Science, Air Force Engineering University, Xi’an, Shaanxi, P. R. China e-mail: (see \(kelinglv@163.com,yangruipan@aliyun.com, liu\_yang10.163.com, qiangfu2024@163.com, gmguo_xjtukgd@yeah.net\)).↩︎

2. Manuscript received March 19, 2026; revised    , 2026.↩︎