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We give a geometric description of binary quantum stabilizer codes. In the case of distance \(d=4\) this leads to the notion of a quaternary quantum cap. We describe several recursive constructions for quantum caps, determine the quantum caps in \(PG(3,4)\) and the cardinalities of quantum caps in \(PG(4,4).\)

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The research of Y. Edel takes place within the project “Linear codes and cryptography” of the Research Foundation—Flanders (FWO) (Project nr. G.0317.06), and is supported by the Interuniversitary Attraction Poles Programme—Belgian State—Belgian Science Policy: project P6/26-Bcrypt. The research of J. Bierbrauer was partially supported by NSA Grant H98230-10-1-0159. This work was also partially supported by the Italian Ministero dell’Istruzione, dell’Università e della Ricerca (MIUR) and by the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni (GNSAGA).

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Correspondence to Jürgen Bierbrauer.

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Communicated by S. Ball.

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Bierbrauer, J., Bartoli, D., Faina, G. et al. The structure of quaternary quantum caps. Des. Codes Cryptogr. 72, 733–747 (2014).

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