Quantum codes and optimal pure quantum $(r,δ)$ -LRCs via the MP constructionQuantum codes and optimal pure quantum $(r,δ)$-LRCs via the MP construction

Meng Cao, Kun Zhou

In this paper, we employ MP codes whose defining matrices are $τ$ -optimal defining ( $τ$ -OD) matrices to construct new quantum codes and quantum $(r,δ)$ -LRCs. Specifically, we report the following results: We establish a unified $τ$ -monomial decomposition theorem for invertible self-adjoint matrices over finite fields of arbitrary characteristic, which generalizes the result in "Quantum codes using the $τ$ -OD MP construction" where the characteristic was required to be odd. Based on this theorem, we prove the existence of $τ$ -OD matrices over $\mathbb{F}_{q^2}$ for any characteristic and demonstrate that there exist several new infinite families of $τ$ -OD matrices over $\mathbb{F}_{q^2}$ of characteristic $2$ . As an application of MP codes involving $τ$ -OD matrices, we construct several infinite families of quantum codes with flexible parameters. Within this framework, we present $222$ record-breaking quantum codes that surpass the best-known records maintained in Grassl's database. We propose two effective schemes for constructing optimal pure quantum $(r,δ)$ -LRCs via MP codes. Accordingly, we construct four new infinite families of optimal pure quantum $(r,δ)$ -LRCs with flexible parameters. Notably, we report an interesting phenomenon by exhibiting $30$ optimal pure quantum $(r,δ)$ -LRCs derived from our framework; that is, there exist quantum codes that are not only optimal pure quantum $(r,δ)$ -LRCs but also, according to Grassl's database, best-known, optimal, or record-breaking quantum codes. To the best of our knowledge, the new discovery that quantum codes are simultaneously optimal pure quantum $(r,δ)$ -LRCs and record-breaking quantum codes has not been previously reported in the literature.In this paper, we employ MP codes whose defining matrices are $τ$-optimal defining ($τ$-OD) matrices to construct new quantum codes and quantum $(r,δ)$-LRCs. Specifically, we report the following results: We establish a unified $τ$-monomial decomposition theorem for invertible self-adjoint matrices over finite fields of arbitrary characteristic, which generalizes the result in "Quantum codes using the $τ$-OD MP construction" where the characteristic was required to be odd. Based on this theorem, we prove the existence of $τ$-OD matrices over $\mathbb{F}_{q^2}$ for any characteristic and demonstrate that there exist several new infinite families of $τ$-OD matrices over $\mathbb{F}_{q^2}$ of characteristic $2$. As an application of MP codes involving $τ$-OD matrices, we construct several infinite families of quantum codes with flexible parameters. Within this framework, we present $222$ record-breaking quantum codes that surpass the best-known records maintained in Grassl's database. We propose two effective schemes for constructing optimal pure quantum $(r,δ)$-LRCs via MP codes. Accordingly, we construct four new infinite families of optimal pure quantum $(r,δ)$-LRCs with flexible parameters. Notably, we report an interesting phenomenon by exhibiting $30$ optimal pure quantum $(r,δ)$-LRCs derived from our framework; that is, there exist quantum codes that are not only optimal pure quantum $(r,δ)$-LRCs but also, according to Grassl's database, best-known, optimal, or record-breaking quantum codes. To the best of our knowledge, the new discovery that quantum codes are simultaneously optimal pure quantum $(r,δ)$-LRCs and record-breaking quantum codes has not been previously reported in the literature.

quant-pharXiv:2606.14253Jun 15, 20260 citations

$\sigma$ -OD MP codes and their applications to AQEC codes

Xiaoyan Zhang

Quantum Information ProcessingMay 9, 20260 citations

Constructions of AEAQEC codes via matrix-product codes

Xiusheng Liu, Peng Hu

Discrete MathematicsJul 26, 20245 citations

Quantum Codes Using the $\tau$ -OD MP Construction

Meng Cao, Kun Zhou

In this paper, we propose an effective method called the τ-optimal defining ( τ-OD) matrix-product (MP) construction to derive infinite families of quantum codes with flexible parameters. Through this scheme, we present 100 record-breaking quantum codes, which exceed the best-known lower bounds on the minimum distances of quantum codes listed in Grassl’s online database. To this end, we need to construct infinite families of τ-OD matrices, a subclass of non-singular by columns (NSC) matrices. We begin by presenting two key results concerning matrices over general fields F with characteristic char(F) ≠ 2, which involve the construction of self-adjoint τ-monomial matrices from any invertible self-adjoint matrix. The finite field version of these results not only reveals the existence of τ-OD matrices over Fq2for some permutation τ but also provides a general method for constructing τ-OD matrices over Fq2for any permutation τ. We show that there exist several infinite families of τ-OD matrices over Fq2where the permutations τ can be easily expressed. By utilizing these constructed τ-OD matrices, we present several infinite families of quantum codes with flexible parameters, many of which are record-breaking.In this paper, we propose an effective method called the τ-optimal defining ( τ-OD) matrix-product (MP) construction to derive infinite families of quantum codes with flexible parameters. Through this scheme, we present 100 record-breaking quantum codes, which exceed the best-known lower bounds on the minimum distances of quantum codes listed in Grassl’s online database. To this end, we need to construct infinite families of τ-OD matrices, a subclass of non-singular by columns (NSC) matrices. We begin by presenting two key results concerning matrices over general fields F with characteristic char(F) ≠ 2, which involve the construction of self-adjoint τ-monomial matrices from any invertible self-adjoint matrix. The finite field version of these results not only reveals the existence of τ-OD matrices over Fq2for some permutation τ but also provides a general method for constructing τ-OD matrices over Fq2for any permutation τ. We show that there exist several infinite families of τ-OD matrices over Fq2where the permutations τ can be easily expressed. By utilizing these constructed τ-OD matrices, we present several infinite families of quantum codes with flexible parameters, many of which are record-breaking.

IEEE Transactions on Information TheoryNov 17, 20254 citations

Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$

Hai Quang Dinh, Bac Trong Nguyen, Hiep L. Thi +2 more

In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits $p^{s}$ over $\mathbb{F}_{p^{m}}$ . When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.

EPJ Quantum TechnologyJun 17, 20260 citations

Quantum codes and optimal pure quantum

(r,δ)

-LRCs via the MP construction

In this paper, we employ MP codes whose defining matrices are

τ

-optimal defining (

τ

-OD) matrices to construct new quantum codes and quantum

(r,δ)

-LRCs. Specifically, we report the following results: We establish a unified

τ

-monomial decomposition theorem for invertible self-adjoint matrices over finite fields of arbitrary characteristic, which generalizes the result in "Quantum codes using the

τ

-OD MP construction" where the characteristic was required to be odd. Based on this theorem, we prove the existence of

τ

-OD matrices over

\mathbb{F}_{q^2}

for any characteristic and demonstrate that there exist several new infinite families of

τ

-OD matrices over

\mathbb{F}_{q^2}

of characteristic

2

. As an application of MP codes involving

τ

-OD matrices, we construct several infinite families of quantum codes with flexible parameters. Within this framework, we present

222

record-breaking quantum codes that surpass the best-known records maintained in Grassl's database. We propose two effective schemes for constructing optimal pure quantum

(r,δ)

-LRCs via MP codes. Accordingly, we construct four new infinite families of optimal pure quantum

(r,δ)

-LRCs with flexible parameters. Notably, we report an interesting phenomenon by exhibiting

30

optimal pure quantum

(r,δ)

-LRCs derived from our framework; that is, there exist quantum codes that are not only optimal pure quantum

(r,δ)

-LRCs but also, according to Grassl's database, best-known, optimal, or record-breaking quantum codes. To the best of our knowledge, the new discovery that quantum codes are simultaneously optimal pure quantum

(r,δ)

-LRCs and record-breaking quantum codes has not been previously reported in the literature.

Asymmetric entanglement-assisted quantum error-correcting MDS codes of physical qubits

p^{s}

over

\mathbb{F}_{p^{m}}

In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of physical qubits

p^{s}

over

\mathbb{F}_{p^{m}}

. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. If we fix the physical qubits and the logical qubits of EAQEC codes, many of them have larger quantum distances than the well known EAQEC codes in the literature. Moreover, some of our EAQEC codes achieve larger quantum distances while requiring fewer or the same number of entangled bits compared with existing EAQEC codes when the numbers of physical qubits and logical qubits are fixed. We construct asymmetric entanglement-assisted quantum error-correcting (AEAQEC) MDS codes of physical qubits

p^{s}

over

\mathbb{F}_{p^{m}}

. When the physical qubits of the AEAQEC codes is fixed, many of our AEAQEC codes have a significantly larger number of quantum distances compared to all previously constructed AEAQEC codes. We also provide a version of Singleton bound for all AEAQEC codes.