Abstract
Ternary self-orthogonal codes with dual distance three and ternary quantum codes of distance three constructed from ternary self-orthogonal codes are discussed in this paper. Firstly, for given code length n ≥ 8, a ternary [n, k]3 self-orthogonal code with minimal dimension k and dual distance three is constructed. Secondly, for each n ≥ 8, two nested ternary self-orthogonal codes with dual distance two and three are constructed, and consequently ternary quantum code of length n and distance three is constructed via Steane construction. Almost all of these quantum codes constructed via Steane construction are optimal or near optimal, and some of these quantum codes are better than those known before.
Similar content being viewed by others
References
Ashikhim A., Knill E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065–3072 (2001)
Bierbrauer J.: The spectrum of stabilizer quantum codes of distance 3. http://www.math.mtu.edu/~jbierbra/. Accessed May 2010.
Bierbrauer J., Edel Y.: Quantum twisted codes. J. Comb. Des. 8, 174–188 (2000)
Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78, 405–409 (1997)
Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error-correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)
Feng K.: Quantum codes [[6, 2, 3]] p and [[7, 3, 3]] p (p ≥ 3) exist. IEEE Trans. Inf. Theory 48, 2384–2391 (2002).
Gottesman D.: Stabilizer codes and quantum error correction. PhD Thesis, California Institute of Technology, quant-ph/9707027 (1997).
Grassl M., Beth T., Rotteler M.: On optimal quantum codes. Int. J. Quantum Inf. 2, 55–65 (2004)
Hamada M.: Concatenated quantum codes constructible in polynomial time: efficient decoding and error correction. IEEE Trans. Inf. Theory 54, 5689–5715 (2008)
Hu D., Tang W., Zhao M. et al.: Graphical nonbinary quantum error-correcting codes. Phys. Rev. A 78, 012306 (2008)
Huffman W.C., Pless V.: Fundamentals of Error Correcting Codes. Cambridge University Press, Cambridge (2003)
Ketkar A., Klappenecker A., Kumar S.: Nonbinary stablizer codes over finite fields. IEEE Trans. Inf. Theory 52, 4892–4914 (2006)
Li R., Li X.: Binary construction of quantum codes of minimum distance three and four. IEEE Trans. Inf. Theory 50, 1331–1336 (2004)
Liu J.: Ternary quantum codes of minimum distance three. Int. J. Quantum Inf. 8, 1179–1186 (2010)
Mallows C.L., Pless V., Sloane N.J.A.: Self-dual codes over GF(3). SIAM J. Appl. Math. 31, 649–666 (1976)
Rains E.M.: Nonbinary quantum codes. IEEE Trans. Inf. Theory 45, 1827–1832 (1999)
Rains E.M., Sloane N.J.A.: Self-dual codes. Handb. Coding Theory 1, 177–294 (1998)
Shor P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995)
Steane A.M.: Error correcting codes in quantum theory. Phys. Rev. Lett. 77, 793–797 (1996a)
Steane A.M.: Simple quantum error correcting codes. Phys. Rev. Lett. 77, C793–C797 (1996b)
Steane A.M.: Enlargement of Calderbank–Shor–Steane quantum codes. IEEE Trans. Inf. Theory 45, 2492–2495 (1999)
Wan Z.: Geometry of Classical Groups over Finite Fields. Studentlitteratur Press, Lund (1993)
Yu S., Dong Y., Chen Q., Oh C.H.: All the optimal stabilizer codes of distance 3. arXiv:0901.1968 (2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Bierbrauer.
Rights and permissions
About this article
Cite this article
Chen, G., Li, R. Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three. Des. Codes Cryptogr. 69, 53–63 (2013). https://doi.org/10.1007/s10623-012-9620-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-012-9620-7