Classical Codes and Quantum Codes Involving the σ Inner Product

In 2019, Carlet et al. introduced the concept of $\sigma $ duals of linear codes involving the $\sigma $ inner product, which generalizes the Euclidean, Hermitian and $\ell $ -Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the $\sigma $ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the $\sigma $ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing $\sigma $ self-orthogonal (SO) and $\sigma $ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the $\sigma $ 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 $\sigma $ 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 $(q^{2}-1)(q+2)$ and $q^{2}(q+2)$ , 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.