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

• Published in: PloS one Vol. 20; no. 6; p. e0325027
• Main Authors: He, Xianmang, Wang, Jingli, Huang, Chunfang, Chen, Yindong
• Format: Journal Article
• Language: English
• Published: United States Public Library of Science 05.06.2025; Public Library of Science (PLoS)
• Subjects: Algorithms; Codes; Coding theory; Computer and Information Sciences; Construction; Error correction & detection; Integers; Methods; Physical Sciences; Polynomials; Quantum Theory; Reed-Solomon codes; Research and Analysis Methods; China
• ISSN: 1932-6203; 1932-6203
• DOI: 10.1371/journal.pone.0325027

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 m 1 and m 2 , 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 .