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

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.

Jie Liu, Peng Hu, Xiusheng Liu

Jie Liu, Xiusheng Liu

Xiusheng Liu, Peng Hu

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.

Hui Li, Xiusheng Liu