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arXiv · 1502.05267

Quantum MDS Codes over Small Fields

Markus Grassl·Martin Roetteler
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Abstract

We consider quantum MDS (QMDS) codes for quantum systems of dimension $q$ with lengths up to $q^2+2$ and minimum distances up to $q+1$. We show how starting from QMDS codes of length $q^2+1$ 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 $n_0(q,d)$ on, are achievable by shortening. Some additional codes that fill gaps in the list of achievable lengths are presented as well along with a construction of a family of QMDS codes of length $q^2+2$, where $q=2^m$, that appears to be new.

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Markus Grassl, Martin Roetteler. 2015-02-18. Quantum MDS Codes over Small Fields. https://doi.org/10.1109/isit.2015.7282626

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