Exploring new lengths for q-ary quantum MDS codes with larger distance

Abstract

In the past decade, the construction of quantum maximum distance separable codes (MDS for short) has been extensively studied. For the length n=q2−1m, 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 n=q2−1m, 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 (m1,m2), where m=m1×m2m1+m2−2 is an integer, with the additional requirement that the greatest common divisor (gcd) of m with both m1 and m2, gcd(m,m1)>1 and gcd(m,m2)>1, and gcd(m1,m2)=2. The quantum MDS codes presented herein are novel and exhibit distance parameters exceeding q2.