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}_q² for any characteristic and demonstrate that there exist several new infinite families of τ-OD matrices over mathbb{F}_q² 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.