Innovations in Incidence Geometry

The geometry of quantum codes

Jürgen Bierbrauer, Giorgio Faina, Massimo Giulietti, Stefano Marcugini, and Fernanda Pambianco

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We give a geometric interpretation of additive quantum stabilizer codes in terms of sets of lines in binary symplectic space. It is used to obtain synthetic geometric constructions and non-existence results. In particular several open problems are removed from Grassl’s database.

Article information

Innov. Incidence Geom., Volume 6-7, Number 1 (2007), 53-71.

Received: 29 February 2008
Accepted: 14 May 2008
First available in Project Euclid: 28 February 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 51E21: Blocking sets, ovals, k-arcs 94B05: Linear codes, general 94B27: Geometric methods (including applications of algebraic geometry) [See also 11T71, 14G50] 94B60: Other types of codes

quantum codes symplectic geometry Hermitian form additive codes Blokhuis-Brouwer construction APN function spread quantum cap


Bierbrauer, Jürgen; Faina, Giorgio; Giulietti, Massimo; Marcugini, Stefano; Pambianco, Fernanda. The geometry of quantum codes. Innov. Incidence Geom. 6-7 (2007), no. 1, 53--71. doi:10.2140/iig.2008.6.53.

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  • J. Bierbrauer, Introduction to Coding Theory, Chapman and Hall, CRC Press, 2004.
  • J. Bierbrauer and Y. Edel, Quantum twisted codes, J. Combin. Des. 8 (2000), 174–188.
  • J. Bierbrauer, G. Faina, S. Marcugini and F. Pambianco, Additive quaternary codes of small length, Proceedings ACCT, Zvenigorod (Russia) September 2006, 15–18.
  • J. Bierbrauer, Y. Edel, G. Faina, S. Marcugini and F. Pambianco, Short additive quaternary codes, IEEE Trans. Inform. Theory, to appear.
  • A. Blokhuis and A. E. Brouwer, Small additive quaternary codes, European J. Combin. 25 (2004), 161–167.
  • A. R. Calderbank, E. M. Rains, P. M. Shor and N. J. A. Sloane, Quantum error-correction via codes over $\GF(4)$, IEEE Trans. Inform. Theory 44 (1998), 1369–1387.
  • C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr. 15 (1998), 125–156.
  • Y. Edel,$\sim$yves.
  • Y. Edel and J. Bierbrauer, $41$ is the largest size of a cap in $\PG(4,4)$, Des. Codes Cryptogr. 16(1999), 151–160.
  • ––––, Large caps in small spaces, Des. Codes Cryptogr. 23 (2001), 197–212.
  • ––––, The largest cap in $\mathsf{AG}(4,4)$ and its uniqueness, Des. Codes Cryptogr. 29 (2003), 99–104.
  • D. Glynn, A $126$-cap in $\PG(5,4)$ and its corresponding $[126,6,88]$-code, Util. Math. 55 (1999), 201–210.
  • M. Grassl,