Results in Mathematics

, Volume 68, Issue 3–4, pp 345–359 | Cite as

Codes Over a Subset of Octonion Integers



In this paper, we define codes over a subset of Octonion integers. We prove that, under certain circumstances, these codes can correct up to two errors for a transmitted vector and the code rate of the codes is greater than the code rate of the codes defined on Quaternion integers.

Mathematics Subject Classification

94B15 94B05 


Block codes cyclic codes integer codes codes over Gaussian integers 


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  1. 1.
    Bales J.W.: A tree for computing the Cayley–Dickson Twist. Mo. J. Math. Sci. 21(2), 83–93 (2009MATHMathSciNetGoogle Scholar
  2. 2.
    Conway, J.H., Smith, D.A.: On Quaternions and Octonions. A.K. Peters, Natick (2003)Google Scholar
  3. 3.
    Cox D.: Primes of the Form x 2 + ny 2: Fermat, Class Field Theory and Complex Multiplication. Wiley, New York (1989)MATHGoogle Scholar
  4. 4.
    Davidoff G., Sarnak P., Valette A.: Elementary Number Theory, Group Theory, and Ramanujan Graphs. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  5. 5.
    Ghaboussi, F., Freudenberger, J.: Codes over Gaussian integer rings. In: Proceedings of 18th Telecommunications forum TELFOR, pp. 662–665 (2010)Google Scholar
  6. 6.
    González Sarabia M., Nava Lara J., Rentería Márquez C., Sarmiento Rosales E.: Parameterized codes over cycles. An. Şt. Univ. Ovid. Constanţa 21(3), 241–255 (2013)MATHGoogle Scholar
  7. 7.
    Güzeltepe: Codes over Hurwitz integers. Discret. Math. 313(5), 704–714 (2013)MATHCrossRefGoogle Scholar
  8. 8.
    Huber K.: Codes over Gaussian integers. IEEE Trans. Inform. Theory 40, 207–216 (1994)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kostadinov H., Morita H., Iijima N., Han Vinck A.J., Manev N.: Soft decoding of integer codes and their application to coded modulation. IEICE Trans. Fundam. E 39(7), 1363–1370 (2010)CrossRefGoogle Scholar
  10. 10.
    Ling S., Xing C.: Coding Theory A First Course. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  11. 11.
    Martinez C., Beivide R., Gabidulin E.: Perfect codes from Cayley graphs over Lipschitz integers. IEEE Trans. Inform. Theory 55(8), 3552–3562 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Morita, H., Han Vinck, A.J., Kostadinov, H.: On soft decoding of coded QAM using integer codes. In: Proceedings of International Symposium on Information Theory and its Applications, ISITA 2004, Parma, Italy, pp. 1321–1325Google Scholar
  13. 13.
    da Nóbrega Neto T.P., Interlando J.C., Favareto M.O., Elia M., Palazzo R. Jr: Lattice constellation and codes from quadratic number fields. IEEE Trans. Inform. Theory 47(4), 1514–1527 (2001)Google Scholar
  14. 14.
    Nishimura S., Hiramatsu T.: A generalization of the Lee distance and error correcting codes. Discret. Appl. Math. 156, 588–595 (2008)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Rifà J.: Groups of complex integer used as QAM signals. IEEE Trans. Inf. Theory 41(5), 1512–1517 (1995)MATHCrossRefGoogle Scholar
  16. 16.
    Savin D.: About some split central simple algebras. An. Şt. Univ. Ovid. Constanţa 22(1), 263–272 (2014)MathSciNetGoogle Scholar
  17. 17.
    Schafer R.D.: An Introduction to Nonassociative Algebras. Academic Press, New York (1966)MATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceOvidius UniversityConstanţaRomania

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