We study the Hermitian hulls of matrix product codes and use them to design linear codes with arbitrary hull dimensions. We continue by looking into some propagation rules that preserve the hull dimensions of $\mathbb{F}_{4}$ -linear codes. We propose a recursive method to keep the Hermitian hull dimension fixed while increasing the dimensions or the minimum distances of codes built from a given $\mathbb{F}_{4}$ -linear code. Using the Hermitian construction route, we derive good entanglement-assisted quantum codes from matrix product codes and recursively iterated codes.We study the Hermitian hulls of matrix product codes and use them to design linear codes with arbitrary hull dimensions. We continue by looking into some propagation rules that preserve the hull dimensions of$\mathbb{F}_{4}$-linear codes. We propose a recursive method to keep the Hermitian hull dimension fixed while increasing the dimensions or the minimum distances of codes built from a given$\mathbb{F}_{4}$-linear code. Using the Hermitian construction route, we derive good entanglement-assisted quantum codes from matrix product codes and recursively iterated codes.

IEEE Information Theory Workshop (ITW)Nov 24, 20242 citations

Entanglement-assisted quantum codes from unitary matrices

Mareth Mam, Martianus Frederic Ezerman, San Ling +1 more

Quantum Information ProcessingJun 13, 20252 citations

Classical Codes and Quantum Codes Involving the σ Inner Product

Meng Cao, Yang Li, Shixin Zhu

In 2019, Carletet al. introduced the concept of σ duals of linear codes involving the σ inner product, which generalizes the Euclidean, Hermitian and ℓ-Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the σ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the σ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing σ self-orthogonal (SO) and σ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the σ construction of quantum stabilizer codes from classical codes. We propose a unified method for constructing two general classes of entanglement-assisted quantum error-correcting codes (EAQECCs) based on the σ hulls of general linear codes. This further yields six types of EAQECCs with flexible parameters based on propagation rules using MP codes under the Euclidean and Hermitian cases. Compared to the best-known ternary EAQECCs, we obtain 17 new ones and 13 of them have improved parameters. Finally, we present two infinite families of q-ary EAQECCs with lengths (q2− 1)(q+ 2) andq2(q+2), respectively. These families include many q-ary QECCs that are not only new according to Grassl’s online database but also surpass those listed in Edel’s online database.In 2019, Carletet al. introduced the concept of σ duals of linear codes involving the σ inner product, which generalizes the Euclidean, Hermitian and ℓ-Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the σ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the σ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing σ self-orthogonal (SO) and σ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the σ construction of quantum stabilizer codes from classical codes. We propose a unified method for constructing two general classes of entanglement-assisted quantum error-correcting codes (EAQECCs) based on the σ hulls of general linear codes. This further yields six types of EAQECCs with flexible parameters based on propagation rules using MP codes under the Euclidean and Hermitian cases. Compared to the best-known ternary EAQECCs, we obtain 17 new ones and 13 of them have improved parameters. Finally, we present two infinite families of q-ary EAQECCs with lengths (q2− 1)(q+ 2) andq2(q+2), respectively. These families include many q-ary QECCs that are not only new according to Grassl’s online database but also surpass those listed in Edel’s online database.

IEEE Transactions on Information TheoryJul 31, 20253 citations

Entanglement-assisted quantum codes from arbitrary binary linear codes

Jianfa Qian, Lina Zhang

Designs Codes and CryptographyJul 25, 201454 citations

\mathbb{F}_{4}

-linear codes. We propose a recursive method to keep the Hermitian hull dimension fixed while increasing the dimensions or the minimum distances of codes built from a given

\mathbb{F}_{4}

-linear code. Using the Hermitian construction route, we derive good entanglement-assisted quantum codes from matrix product codes and recursively iterated codes.