Quantum MDS code with Length .

In the past decade, the construction of quantum maximum distance separable codes (MDS for short) has been extensively studied. For the length , where m is an integer that divides either q  +  1 or q − 1, a complete set of results has been available. In this paper, we dedicate to a previously unexplored cases where the length , subject to the conditions that m is neither a divisor of q − 1 nor q  +  1. Ultimately, this problem can be summarized as exploring the necessary and sufficient conditions for the existence of pairs , where is an integer, with the additional requirement that the greatest common divisor () of m with both m1 and m2, and , and . The quantum MDS codes presented herein are novel and exhibit distance parameters exceeding .