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 [Formula: see text], 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 [Formula: see text], 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 [Formula: see text], where [Formula: see text] is an integer, with the additional requirement that the greatest common divisor ([Formula: see text]) of m with both m1 and m2, [Formula: see text] and [Formula: see text], and [Formula: see text]. The quantum MDS codes presented herein are novel and exhibit distance parameters exceeding [Formula: see text].