Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2018, Cilt: 1 Sayı: 1, 39 - 45, 18.05.2018

Öz

Kaynakça

  • [11] C. Martinez, R. Beivide and E. Gabidulin, Perfect Codes from Cayley Graphs over Lipschitz Integers, IEEE Trans. Inf. Theory, 55 (2009)3552-3562.
  • [12] T. P. da N. Neto, J. C. Interlando., ”Lattice constellation and codes from quadratic number fields,” IEEE Trans. Inform. Theory, vol. 47, No.4, May. 2001.
  • [13] K. Huber., ”Codes Over Gaussian integers,” IEEE Trans. Inform.Theory, vol. 40, pp. 207-216, Jan. 1994.
  • [14] K. Huber., ”Codes Over Eisenstein-Jacobi integers,” AMS. Contemp. Math., vol. 158, pp.165-179, 2004.
  • [15] C. Martinez, R. Beivide and E. Gabidulin., ”Perfect codes for metrics induced by circulant graphs,” IEEE Trans. Inform. Theory, vol. 53, No.9, Sep. 2007.
  • [16] C. Martinez, R. Beivide and E. Gabidulin, ”Perfect Codes from Cayley Graphs over Lipschitz Integers,” IEEE Trans. Inf. Theory, Vol. 55,No. 8, Aug. 2009.
  • [17] G. Davidoff, P. Sarnak, and A. Valette., Elementary Number Theory, Group Theory, and Ramanujan Graphs, Cambridge University Pres,2003.
  • [18] J. H. Conway, D. A. Smith, On Quaternions and Octonions, A K Peters, 2003.
  • [19] M. Guzeltepe, ”The Macwilliams Identity for Lipschitz Weight Enumerators,” GU J Sci. 29(4): 869-877 (2016). ¨
  • [20] M. Guzeltepe, O. Heden, ”Perfect Mannheim, Lipschitz and Hurwitz weight codes”, Math. Communications, Vol. 19 ¨ /2 pp. 253-276, 2014.
  • [1] M. Guzeltepe, Codes over Hurwitz integers, Discrete Mathematics, (2012), doi: 10.1016 ¨ /j.disc.2012.10.020.
  • [21] O. Heden, M. Guzeltepe, ”On perfect 1- error-correcting codes”, Math. Communications, Vol. 20 ¨ /1 pp. 23-35, 2015.
  • [22] O. Heden, M. Guzeltepe, ”Perfect 1-error-correcting Lipschitz weight codes”, Math. Communications, Vol. 21 ¨ /1 pp. 23-30, 2016.
  • [23] M. Guzeltepe, a. Altınel, ”Perfect 1-error-correcting Hurwitz weight codes”, Math. Communications, Vol. 22 ¨ /2 pp. 265-272, 2017.
  • [2] T. Abualrub, ˙ I. S¸iap, Constacyclic codes over F2 + uF2, J. Franklin Inst., 346 (2009) 520-529.
  • [3] ˙ I. S¸iap, T. Abualrub, A. Ghrayeb, Cyclic DNA codes over the ring F2[u]=(u2 − 1) based on the deletion distance, J. Franklin Inst., 346 (2009)731-740.
  • [4] T. Abualrub, ˙ I. S¸iap, Cyclic codes over the rings Z2 + uZ2 and Z2 + uZ2 + u2Z2, Des. Codes Cryptogr., 42 (2007) 273-287.
  • [5] B. Yildiz, S. Karadeniz, Self-dual codesover F2 + uF2 + vF2 + uvF2, J. Franklin Inst., 347 (2010) 1888-1894.
  • [6] B. Yildiz, S. Karadeniz, Cyclic codes over F2 + uF2 + vF2 + uvF2, Des. Codes Cryptogr. 58 (2011) 221-234 DOI 10.1007/s10623-010-9399-3.
  • [7] M. Ozen, M. G ¨ uzeltepe, Cyclic codes over some finite quaternion integer rings, J. Franklin Inst., 348 (2011) 1312-1317. ¨
  • [8] J. H. van Lint, ”Nonexistence theorems for perfect error-correcting codes,” in Computers in Algebra and Number Theory, vol. IV, SIAM-AMSProceedings, 1971.
  • [9] G. Davidoff, P. Sarnak, and A. Valette., Elementary Number Theory, Group Theory, and Ramanujan Graphs, Cambridge University Pres, 2003.[10] J. H. Conway, D. A. Smith, On Quaternions and Octonions, A K Peters, 2003.

On Some Perfect Codes over Hurwitz Integers

Yıl 2018, Cilt: 1 Sayı: 1, 39 - 45, 18.05.2018

Öz

The article considers linear codes over Hurwitz integers. The codes are considered with respect to a new Hurwitz metric. This metric is more suitable for
(QAM)-type constellations than the Hamming Metric and the Lee metric. Also, one error correcting perfect codes with respect to the Hurwitz metric are defined. The decoding algorithm of these codes is obtained. Moreover, a simple comparison in respect to the average energy for the transmitted signal and the bandwidth occupancy is given.

Kaynakça

  • [11] C. Martinez, R. Beivide and E. Gabidulin, Perfect Codes from Cayley Graphs over Lipschitz Integers, IEEE Trans. Inf. Theory, 55 (2009)3552-3562.
  • [12] T. P. da N. Neto, J. C. Interlando., ”Lattice constellation and codes from quadratic number fields,” IEEE Trans. Inform. Theory, vol. 47, No.4, May. 2001.
  • [13] K. Huber., ”Codes Over Gaussian integers,” IEEE Trans. Inform.Theory, vol. 40, pp. 207-216, Jan. 1994.
  • [14] K. Huber., ”Codes Over Eisenstein-Jacobi integers,” AMS. Contemp. Math., vol. 158, pp.165-179, 2004.
  • [15] C. Martinez, R. Beivide and E. Gabidulin., ”Perfect codes for metrics induced by circulant graphs,” IEEE Trans. Inform. Theory, vol. 53, No.9, Sep. 2007.
  • [16] C. Martinez, R. Beivide and E. Gabidulin, ”Perfect Codes from Cayley Graphs over Lipschitz Integers,” IEEE Trans. Inf. Theory, Vol. 55,No. 8, Aug. 2009.
  • [17] G. Davidoff, P. Sarnak, and A. Valette., Elementary Number Theory, Group Theory, and Ramanujan Graphs, Cambridge University Pres,2003.
  • [18] J. H. Conway, D. A. Smith, On Quaternions and Octonions, A K Peters, 2003.
  • [19] M. Guzeltepe, ”The Macwilliams Identity for Lipschitz Weight Enumerators,” GU J Sci. 29(4): 869-877 (2016). ¨
  • [20] M. Guzeltepe, O. Heden, ”Perfect Mannheim, Lipschitz and Hurwitz weight codes”, Math. Communications, Vol. 19 ¨ /2 pp. 253-276, 2014.
  • [1] M. Guzeltepe, Codes over Hurwitz integers, Discrete Mathematics, (2012), doi: 10.1016 ¨ /j.disc.2012.10.020.
  • [21] O. Heden, M. Guzeltepe, ”On perfect 1- error-correcting codes”, Math. Communications, Vol. 20 ¨ /1 pp. 23-35, 2015.
  • [22] O. Heden, M. Guzeltepe, ”Perfect 1-error-correcting Lipschitz weight codes”, Math. Communications, Vol. 21 ¨ /1 pp. 23-30, 2016.
  • [23] M. Guzeltepe, a. Altınel, ”Perfect 1-error-correcting Hurwitz weight codes”, Math. Communications, Vol. 22 ¨ /2 pp. 265-272, 2017.
  • [2] T. Abualrub, ˙ I. S¸iap, Constacyclic codes over F2 + uF2, J. Franklin Inst., 346 (2009) 520-529.
  • [3] ˙ I. S¸iap, T. Abualrub, A. Ghrayeb, Cyclic DNA codes over the ring F2[u]=(u2 − 1) based on the deletion distance, J. Franklin Inst., 346 (2009)731-740.
  • [4] T. Abualrub, ˙ I. S¸iap, Cyclic codes over the rings Z2 + uZ2 and Z2 + uZ2 + u2Z2, Des. Codes Cryptogr., 42 (2007) 273-287.
  • [5] B. Yildiz, S. Karadeniz, Self-dual codesover F2 + uF2 + vF2 + uvF2, J. Franklin Inst., 347 (2010) 1888-1894.
  • [6] B. Yildiz, S. Karadeniz, Cyclic codes over F2 + uF2 + vF2 + uvF2, Des. Codes Cryptogr. 58 (2011) 221-234 DOI 10.1007/s10623-010-9399-3.
  • [7] M. Ozen, M. G ¨ uzeltepe, Cyclic codes over some finite quaternion integer rings, J. Franklin Inst., 348 (2011) 1312-1317. ¨
  • [8] J. H. van Lint, ”Nonexistence theorems for perfect error-correcting codes,” in Computers in Algebra and Number Theory, vol. IV, SIAM-AMSProceedings, 1971.
  • [9] G. Davidoff, P. Sarnak, and A. Valette., Elementary Number Theory, Group Theory, and Ramanujan Graphs, Cambridge University Pres, 2003.[10] J. H. Conway, D. A. Smith, On Quaternions and Octonions, A K Peters, 2003.
Toplam 22 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Articles
Yazarlar

Murat Güzeltepe

Yayımlanma Tarihi 18 Mayıs 2018
Yayımlandığı Sayı Yıl 2018 Cilt: 1 Sayı: 1

Kaynak Göster

APA Güzeltepe, M. (2018). On Some Perfect Codes over Hurwitz Integers. Mathematical Advances in Pure and Applied Sciences, 1(1), 39-45.
AMA Güzeltepe M. On Some Perfect Codes over Hurwitz Integers. MAPAS. Mayıs 2018;1(1):39-45.
Chicago Güzeltepe, Murat. “On Some Perfect Codes over Hurwitz Integers”. Mathematical Advances in Pure and Applied Sciences 1, sy. 1 (Mayıs 2018): 39-45.
EndNote Güzeltepe M (01 Mayıs 2018) On Some Perfect Codes over Hurwitz Integers. Mathematical Advances in Pure and Applied Sciences 1 1 39–45.
IEEE M. Güzeltepe, “On Some Perfect Codes over Hurwitz Integers”, MAPAS, c. 1, sy. 1, ss. 39–45, 2018.
ISNAD Güzeltepe, Murat. “On Some Perfect Codes over Hurwitz Integers”. Mathematical Advances in Pure and Applied Sciences 1/1 (Mayıs 2018), 39-45.
JAMA Güzeltepe M. On Some Perfect Codes over Hurwitz Integers. MAPAS. 2018;1:39–45.
MLA Güzeltepe, Murat. “On Some Perfect Codes over Hurwitz Integers”. Mathematical Advances in Pure and Applied Sciences, c. 1, sy. 1, 2018, ss. 39-45.
Vancouver Güzeltepe M. On Some Perfect Codes over Hurwitz Integers. MAPAS. 2018;1(1):39-45.