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When constructing quantum codes under the entanglement-assisted (EA) stabilizer formalism, one can ignore the limitation of dual-containing condition. This allows us to construct EA quantum error-correcting codes (QECCs) from any classical linear codes. The main contribution of this manuscript is to make a general statement for determining the optimal number of pre-shared qubits instead of presenting only specific cases. Let n = q − 1 2 a (q + 1) and gcd (n , q) = 1 , where q is an odd prime power, 2 a | (q − 1) and a ≥ 3. By deeply investigating the decomposition of the defining set of negacyclic codes, we generalize the number of pre-shared entanglement pairs of Construction (1) in Lu et al. [Quantom Inf. Process. 17 (2018) 69] from c = 2 , 4 to arbitrary even numbers less than or equal to 2 a. Consequently, a series of EA quantum maximum distance separable (EAQMDS) codes can be produced. The absolute majority of them are new and the minimum distance can be up to q + q − 1 2 a . Moreover, this method can be applied to construct many other families of EAQECCs with good parameters, especially large minimum distance. [ABSTRACT FROM AUTHOR]