MDS Constacyclic Codes of Prime Power Lengths Over Finite Fields and Construction of Quantum MDS Codes

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 ps over Fpm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {F}_{p^{m}}$\end{document}. 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.