We consider quantum MDS (QMDS) codes for quantum systems of dimension q with lengths up to q2 + 2 and minimum distances up to q + 1. We show how starting from QMDS codes of length q2 + 1 based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide numerical evidence for our conjecture that almost all admissible lengths, from a lower bound n0(q,d) on, are achievable by shortening. Some additional codes that \ufb01ll gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length q2 +2, where q = 2m, that appears to be new.<\/p>\n","protected":false},"excerpt":{"rendered":"
We consider quantum MDS (QMDS) codes for quantum systems of dimension q with lengths up to q2 + 2 and minimum distances up to q + 1. We show how starting from QMDS codes of length q2 + 1 based on cyclic and constacyclic codes, new QMDS codes can be obtained by shortening. We provide […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"footnotes":""},"msr-content-type":[3],"msr-research-highlight":[],"research-area":[243138],"msr-publication-type":[193716],"msr-product-type":[],"msr-focus-area":[],"msr-platform":[],"msr-download-source":[],"msr-locale":[268875],"msr-field-of-study":[],"msr-conference":[],"msr-journal":[],"msr-impact-theme":[],"msr-pillar":[],"class_list":["post-168515","msr-research-item","type-msr-research-item","status-publish","hentry","msr-research-area-quantum","msr-locale-en_us"],"msr_publishername":"IEEE - Institute of Electrical and Electronics Engineers","msr_edition":"Proceedings of the 2015 IEEE International Symposium on Information Theory (ISIT'15), Hong Kong","msr_affiliation":"","msr_published_date":"2015-06-01","msr_host":"","msr_duration":"","msr_version":"","msr_speaker":"","msr_other_contributors":"","msr_booktitle":"","msr_pages_string":"1104\u20131108","msr_chapter":"","msr_isbn":"","msr_journal":"","msr_volume":"","msr_number":"","msr_editors":"","msr_series":"","msr_issue":"","msr_organization":"","msr_how_published":"","msr_notes":"See also arXiv preprint arXiv:1502.05267","msr_highlight_text":"","msr_release_tracker_id":"","msr_original_fields_of_study":"","msr_download_urls":"","msr_external_url":"","msr_secondary_video_url":"","msr_longbiography":"","msr_microsoftintellectualproperty":1,"msr_main_download":"204324","msr_publicationurl":"","msr_doi":"","msr_publication_uploader":[{"type":"file","title":"Grassl%20Roetteler%20QMDS%201502.05267.pdf","viewUrl":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/Grassl20Roetteler20QMDS201502.05267.pdf","id":204324,"label_id":0}],"msr_related_uploader":"","msr_attachments":[{"id":204324,"url":"https:\/\/www.microsoft.com\/en-us\/research\/wp-content\/uploads\/2016\/02\/Grassl20Roetteler20QMDS201502.05267.pdf"}],"msr-author-ordering":[{"type":"text","value":"M. 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