Markus Grassl - On Quantum MDS Codes

Coding Theory and Cryptography Seminars

11 June 2020, Karan Khathuria, 52 views

Abstract

A quantum error-correcting code (QECC), denoted by ((n,K,d))_q, is a K-dimensional subspace of the complex vector space C^(q^⊗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 log_q(K) ≤ 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 q^2+1 (or q^2+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).

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