Exploring new lengths for <i> q </i> -ary quantum MDS codes with larger distance Exploring new lengths for <i> q </i> -ary quantum MDS codes with larger distance

Abstract

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