PERFECT CODES OVER HURWITZ INTEGERS INDUCED BY CIRCULANT GRAPHS

Year 2022, Volume: 5 Issue: 1, 24 - 35, 01.03.2022

Murat Güzeltepe , Gökhan Güner

https://doi.org/10.33773/jum.985160

Cited By: 1

https://izlik.org/JA62KG44JD

Abstract

In this paper, a new family of t−error correcting perfect codes over Hurwitz integers is
presented. To obtain these perfect codes, the perfect t−dominating sets over the circulant
graphs are used. The codewords of such perfect codes are generated by the elements of a
subgroup of the considered group.

Keywords

Circulant graphs , Hurwitz integers , perfect codes , quadrature amplitude modulation (QAM) constellations

Project Number

116F318

References

• [1] M. Güzeltepe, Codes over Hurwitz integers, Discrete Math. 313(2013), 704–714.
• [2] M. Güzeltepe, A. Altınel, Perfect 1−error-correcting Hurwitz weight codes, Math. Commun. 22(2017), 265–272.
• [3] M. Güzeltepe, O. Heden, Perfect Mannheim, Lipschitz and Hurwitz weight codes, Math. Commun. 19(2014), 253–276.
• [4] R.W. Hamming, Error detecting and error correcting codes, Bell System Technical Journal 29(1950), 147—160.
• [5] O. Heden, A new construction of group and nongroup perfect codes, Information and Control 34(1977), 314-–323.
• [6] O. Heden, M. Güzeltepe, On perfect 1-ε-error-correcting codes, Math. Commun. 20(2015), 23—35.
• [7] O. Heden, M. Güzeltepe, Perfect 1−error-correcting Lipschitz weight codes, Math. Commun. 21(2016), 23-–30.
• [8] K. Huber, Codes over Gaussian integers, IEEE Trans. Inform. Theory 40(1994), 207—216.
• [9] C.Y. Lee, Some properties of non-binary error correcting codes, IEEE Trans. Inform. Theory 4(1958), 77—82.
• [10] B. B. Lindström , On group and nongroup perfect codes in q symbols, Math. Scand. 25(1969), 149-–158.
• [11] C. Martínez, E. Stafford, R. Beivide, E. Gabidulin, Perfect codes over Lipschitz integers, in: Proc. IEEE Int. Symp. Information Theory, Nice, 2007, 1366— 1370.
• [12] C. Martínez, R. Beivide, E. Gabidulin, Perfect codes from Cayley graphs over Lipschitz integers, IEEE Trans. Inf. Theory 55(2009), 3552-–3562.
• [13] C. Martínez, R. Beivide, E. Gabidulin, Perfect codes for metrics induced by circulant graphs, IEEE Trans. Inf. Theory 53(2007), 3042—3052.
• [14] J. Schönheim, On linear and nonlinear, single-error-correcting q−nary perfect codes, Information and Control 12(1968), 23-–26.
• [15] Y.L. Vasil’ev, On nongroup close-packed codes, Problemi Tekhn. Kibernet. Robot. 8(1962), 337—339.
• [16] J. G. Proakis, M. Salehi, Communications Systems Engineering, Second Edition, Prentice Hall.
• [17] S. Lin, D. J. Costello, Jr., Error Control Coding, Second Edition, Prentice Hall.