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Quantum Codes and Abelian Subgroups of the Extra-Special Group

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Abstract

A method for constructing a new class of quantum codes is proposed. In the method, properties of the extra-special matrix group are exploited. This class of codes is wider than that considered in [1–4] (CSS-codes). In particular, this class includes the one-error-correcting quantum “Hamming code” (which is not a CSS-code) of length n = 2m, the number of elements in it being close to the maximum possible. The latter result is one of the main results of the paper.

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REFERENCES

  1. Steane, A.M., Error Correcting Codes in Quantum Theory, Phys. Rev. Letters, 1996, vol. 77, pp. 793–797.

    Google Scholar 

  2. Steane, A.M., Enlargement of Calderbank-Shor-Steane Codes, IEEE Trans. Inf. Theory, 1999, vol. 45, no. 7, pp. 2492–2495.

    Google Scholar 

  3. Calderbank, A.R. and Shor, P.W., Good Quantum Error-Correcting Codes Exist, Phys. Rev, 1996, vol. 54, no. 2, pp. 1098–1105.

    Google Scholar 

  4. Calderbank, A.R., Rains, E.M., Shor, P.W., and Sloane, N.J.A., Quantum Errors Correction via Codes over GF(4), IEEE Trans. Inf. Theory, 1998, vol. 44, pp. 1369–1387.

    Google Scholar 

  5. Cohen, G., Encheva, S., and Litsyn, S., On Binary Construction Quantum Codes, IEEE Trans. Inf. Theory, 1999, vol. 45, no. 7, pp. 2495–2509.

    Google Scholar 

  6. Gottesman, D., A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound, Phys. Rev. A, 1996, vol. 54, pp. 1862–1868.

    Google Scholar 

  7. Ashikhmin, A. and Litsyn, S., Upper Bounds on the Size of Quantum Code, IEEE Trans. Inf. Theory, 1999, vol. 45, no. 4, pp. 1206–1215.

    Google Scholar 

  8. Rains, E.M., Quantum Weight Enumerators, IEEE Trans. Inf. Theory, 1998, vol. 44, no. 4, pp. 1388–1394.

    Google Scholar 

  9. Knill, E. and Laamme, R., A Theory of Quantum Error Correcting Codes, 1997, Phys. Rev. A, vol. 55, pp. 900–911.

  10. Kostrikin, A.I., Vvedenie v algebru, Moscow: Nauka, 1977. Translated under the title Introduction to Algebra, New York: Springer, 1982.

    Google Scholar 

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Authors
  1. V. M. Sidelnikov
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Sidelnikov, V.M. Quantum Codes and Abelian Subgroups of the Extra-Special Group. Problems of Information Transmission 38, 194–202 (2002). https://doi.org/10.1023/A:1020361118261

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