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Year 2019, Volume: 37 Issue: 4, 1325 - 1333, 01.12.2019

Abstract

References

  • [1] Andrews G. E., (1994) Number Theory, first ed., Dover Publications, Inc., New York,
  • [2] Bonnecaze A., Solé P. and Calderbank A. R., (1995) Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 2, 41,366–377.
  • [3] Heden, O., (2011) The non-existence of some perfect codes over non-prime power alphabets, Discrete Math. 311, 14, 1344–1348.
  • [4] Horak P., (2009) On perfect Lee codes, Discrete Math. 309, 18, 5551–5561.
  • [5] Jain, S., Nam K. B. and Lee K. S. (2005) On some perfect codes with respect to Lee metric, Linear Algebra Appl. 405, 104–120.
  • [6] Lee, C.Y., (1958) Some properties of non-binary error correcting codes, IEEE Trans. Information Theory 4, 2, 77–82.
  • [7] van Lint J. H., (1975) A survey of perfect codes, Rocky Mountain J. Math. 5, 199–226.
  • [8] Malyugin S. A., (2004) On a lower bound on the number of perfect binary codes, Discrete Appl. Math. 135,1-3, 157–160.
  • [9] Özen M. and Şiap V., (2012) On the existence of perfect linear codes over Z_4 with respect to homogenous weight, Appl. Math. Sci. 6 no. 41, 2005–2011.
  • [10] Siap I., Özen M. and Şiap V., (2013) On the existence of perfect linear codes over Z_(2l) with respect to homogenous metric, Arabian Journal for Science and Engineering 38 ,8, 2189–2192.
  • [11] Tietäväinen A., (1973), On the nonexistence of perfect codes over finite fields, SIAM J. Appl. Math. 24 ,1, 88–96.
  • [12] Ungerboeck G., (1982) Channel coding with multilevel/phase signals, IEEE Trans. Information Theory 28,1, 55–67.
  • [13] Rosenbloom M.Yu. and Tsfasman M.A., (1997) Codes for m-metric, Problems of Information Transmission 33 ,45–52.

ON PERFECT CODES IN THE LEE-ROSENBLOOM-TSFASMAN-JAIN METRIC

Year 2019, Volume: 37 Issue: 4, 1325 - 1333, 01.12.2019

Abstract

In this paper, we study perfect codes in the Lee-Rosenbloom-Tsfasman-Jain (LRTJ) metric over the finite field Z_p. We begin by deriving some new upper bounds focusing on the number of parity check digits for linear codes correcting all error vectors of LRTJ weight up to w, 1≤w≤4. Furthermore, we establish sufficient conditions for the existence of perfect codes correcting all error vectors with certain weights. We also search for linear codes which attain these bounds to determine the possible parameters of perfect codes. Moreover, we derive parity check matrices corresponding linear codes correcting all error vectors of LRTJ weight 1 over Z_p, and correcting all error vectors of LRTJ weight up to 2 over Z_3 and Z_11. We also construct perfect codes for these cases. Lastly, we obtain non-existence results on w-perfect linear codes over Z_p for 2≤w≤4.

References

  • [1] Andrews G. E., (1994) Number Theory, first ed., Dover Publications, Inc., New York,
  • [2] Bonnecaze A., Solé P. and Calderbank A. R., (1995) Quaternary quadratic residue codes and unimodular lattices, IEEE Trans. Inform. Theory 2, 41,366–377.
  • [3] Heden, O., (2011) The non-existence of some perfect codes over non-prime power alphabets, Discrete Math. 311, 14, 1344–1348.
  • [4] Horak P., (2009) On perfect Lee codes, Discrete Math. 309, 18, 5551–5561.
  • [5] Jain, S., Nam K. B. and Lee K. S. (2005) On some perfect codes with respect to Lee metric, Linear Algebra Appl. 405, 104–120.
  • [6] Lee, C.Y., (1958) Some properties of non-binary error correcting codes, IEEE Trans. Information Theory 4, 2, 77–82.
  • [7] van Lint J. H., (1975) A survey of perfect codes, Rocky Mountain J. Math. 5, 199–226.
  • [8] Malyugin S. A., (2004) On a lower bound on the number of perfect binary codes, Discrete Appl. Math. 135,1-3, 157–160.
  • [9] Özen M. and Şiap V., (2012) On the existence of perfect linear codes over Z_4 with respect to homogenous weight, Appl. Math. Sci. 6 no. 41, 2005–2011.
  • [10] Siap I., Özen M. and Şiap V., (2013) On the existence of perfect linear codes over Z_(2l) with respect to homogenous metric, Arabian Journal for Science and Engineering 38 ,8, 2189–2192.
  • [11] Tietäväinen A., (1973), On the nonexistence of perfect codes over finite fields, SIAM J. Appl. Math. 24 ,1, 88–96.
  • [12] Ungerboeck G., (1982) Channel coding with multilevel/phase signals, IEEE Trans. Information Theory 28,1, 55–67.
  • [13] Rosenbloom M.Yu. and Tsfasman M.A., (1997) Codes for m-metric, Problems of Information Transmission 33 ,45–52.
There are 13 citations in total.

Details

Primary Language English
Journal Section Research Articles
Authors

Merve Bulut Yılgör 0000-0001-6842-1109

Fatih Demirkale This is me 0000-0002-6114-1291

Publication Date December 1, 2019
Submission Date June 18, 2019
Published in Issue Year 2019 Volume: 37 Issue: 4

Cite

Vancouver Bulut Yılgör M, Demirkale F. ON PERFECT CODES IN THE LEE-ROSENBLOOM-TSFASMAN-JAIN METRIC. SIGMA. 2019;37(4):1325-33.

IMPORTANT NOTE: JOURNAL SUBMISSION LINK https://eds.yildiz.edu.tr/sigma/