MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes
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MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes Hai Q. Dinh1,2 · Ramy Taki ElDin3 · Bac T. Nguyen4
· Roengchai Tansuchat5
Received: 14 April 2020 / Accepted: 22 June 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract If we fix the code length n and dimension k, maximum distance separable (briefly, MDS) codes form an important class of codes because the class of MDS codes has the greatest error-correcting and detecting capabilities. In this paper, we establish all MDS constacyclic codes of length p s over Fpm . We also give some examples of MDS constacyclic codes over finite fields. As an application, we construct all quantum MDS codes from repeated-root codes of prime power lengths over finite fields using the CSS and Hermitian constructions. We provide all quantum MDS codes constructed from dual codes of repeated-root codes of prime power lengths over finite fields using the Hermitian construction. They are new in the sense that their parameters are different from all the previous constructions. Moreover, some of them have larger Hamming distances than the well known quantum error-correcting codes in the literature. Keywords Constacyclic codes · Hamming distance · MDS codes · Quantum MDS codes
Bac T. Nguyen
[email protected] Hai Q. Dinh [email protected] Ramy Taki ElDin [email protected] Roengchai Tansuchat [email protected] 1
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
Faculty of Engineering, Ain Shams University, Cairo, Egypt
4
Department of Basic Sciences, Thai Nguyen University of Economics and Business Administration, Thai Nguyen, Thai Nguyen Province, Vietnam
5
Centre of Excellence in Econometrics, Faculty of Economics, Chiang Mai University, Chiang Mai, 52000, Thailand
International Journal of Theoretical Physics
1 Introduction Berlekamp [3] first studied constacyclic codes over finite fields, which have a rich algebriac structure and are generalizations of cyclic and negacyclic codes. A -constacyclic code F[x] of length n over F is an ideal of xF[x] n − . If = 1 ( = −1), then an ideal of x n − is called a cyclic (negacyclic) code. Cyclic codes over finite fields were first studied by Prange in 1957 [54]. After that, negacyclic codes over finite fields were considered by Berlekamp [4]. Berman [5] first initiated the case (n, p) = p, where n is the code length and p is the characteristic of the fields. If (n, p) = p, then codes over finite fields are so-called repeated-root codes. Such codes are also studied by some authors (for examples, Massey et al. [52], Roth and Seroussi [56], and van Lint [64]). Recently, Dinh, in a series of papers ([21, 23–26]), determined the algebraic structures of constacyclic codes in terms of generator polynomials over Fpm of length mp s , where m =
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