Talk
eSeminar: On Quantum MDS Codes
Talk by Dr. Markus Grassl
Date: 10.06.20 Time: 16.00 - 17.00 Room:
(**This eSeminar will take place on Zoom, using the same meeting details as previous seminars. If you do not have meeting details, please contact karan.khathuria@math.uzh.ch **)
A quantum error-correcting code (QECC), denoted
by ((n,K,d))q, is a K-dimensional
subspace of the complex vector
space Cq⊗n that is
able to correct up do d-1 erasures. A quantum MDS (QMDS)
code is a code of maximal possible dimension meeting the quantum
Singleton bound logqK ≤ n+2-2d. Most known
QMDS codes are based on Hermitian self-orthogonal classical MDS
codes. It has recently been shown [3] that regardless of the
underlying construction, QMDS codes share many (but not all)
properties with their classical counterparts. The QMDS conjecture
states that the length of nontrivial codes is bounded by
q2+1 (or q2+2 in special
cases). While QMDS codes of maximal length are known for many cases,
it appears to be difficult to find codes of distance d > q+1 (see
[1,2]).
The talk addresses the question of finding QMDS codes in
general and presents a couple of related open questions in algebraic
coding theory.
[1] Ball, Simeon, "Some constructions of quantum MDS codes'', preprint arXiv:1907.04391, (2019).
[2] Grassl, Markus and Roetteler, Martin, "Quantum MDS Codes over Small Fields'', Proceedings 2015 IEEE International Symposium on Information Theory (ISIT 2015), pp. 1104--1108, (2015). DOI: 10.1109/ISIT.2015.7282626, preprint arXiv:1502.05267.
[3] Huber, Felix and Grassl, Markus, "Quantum Codes of Maximal Distance and Highly Entangled Subspaces'', preprint arXiv:1907.07733, (2019).