Plotkin-optimal pure quantum LRCs with large minimum distances from NMDS codes

Quantum locally recoverable codes (qLRCs) are a class of quantum error-correcting codes that can recover an erased qudit by accessing only a few other qudits. This paper develops a general framework for constructing optimal pure qLRCs that attain the Plotkin-like bound, referred to as Plotkin-optimal pure qLRCs, from Hermitian dual-containing near maximum-distance separable (NMDS) codes. Focusing on a special class of NMDS codes called extended subcodes of generalized Reed-Solomon (ESGRS) codes, we determine their localities by analyzing certain subset sum problems over finite fields. By further exploiting the algebraic structures of ESGRS codes and related codes, we construct four families of Plotkin-optimal pure qLRCs with large lengths and more flexible minimum distances than existing constructions. These constructions provide strong global error-correction capability while retaining local recovery. To the best of our knowledge, they give the first families of qLRCs attaining the pure Plotkin-like bound. In addition, by using the resulting locality property, we resolve in the negative a related problem on ESGRS codes supporting combinatorial designs.