19316

An algebraic construction technique for codes over Hurwitz integers

Let $\alpha$ be a prime Hurwitz integer. $\mathcal{H}_{\alpha}$, which is the set of residual class with respect to related modulo function in the rings of Hurwitz integers, is a subset of $\mathcal{H},$ which is the set of all Hurwitz integers. In this study, we present an algebraic construction technique, which is a modulo function formed depending on two modulo operations, for codes over Hurwitz integers. We consider left congruent modulo $\alpha,$ and the domain of related modulo function is $\mathbb{Z}_{N(\alpha)},$ which is residual class ring of ordinary integers with $N(\alpha)$ elements. Therefore, we obtain the residue class rings of Hurwitz integers with $N(\alpha)$ size. In addition, we present some results for mathematical notations used in two modulo functions, and for the algebraic construction technique formed depending upon two modulo functions. Moreover, we presented graphs obtained by graph layout methods, such as spring, high-dimensional, and spiral embedding, for the set of the residual class obtained with respect to the related modulo function in the rings of Hurwitz integers.

Keywords:

algebraic construction, average energy, block code code rate, graph,

PDF

___

[1] J. Freudenberger, F. Ghaboussi and S. Shavgulidze, New coding techniques for codes over Gaussian integers, IEEE Transactions on Communications 61 (8), 3114-3124, 2013.
[2] J. Freudenberger and S. Shavgulidze, New four-dimensional signal constellations from Lipschitz integers for transmission over the Gaussian channel, IEEE Transactions on Communications 63 (7), 2420-2427, 2015.
[3] K. Huber, Codes over Gaussian integers, IEEE Trans. Inform. Theory 40 (1), 207-216, 1994.
[4] K. Huber, Codes over Eisenstein-Jacobi integers, Finite fields: theory, applications, and algorithms 168, 165179, 1994.
[5] M. Güzeltepe, Codes over Hurwitz integers, Discrete Math. 313 (5), 704-714, 2013.
[6] M. Güzeltepe, On some perfect codes over Hurwitz integers, Mathematical Advances in Pure and Applied Sciences 1 (1), 39-45, 2018.
[7] M. Güzeltepe and A. Altınel, Perfect 1-error-correcting Hurwitz weight codes, Math. Commun. 22 (2), 265-272, 2017.
[8] M. Güzeltepe and O. Heden, Perfect Mannheim, Lipschitz and Hurwitz weight codes, Math. Commun. 19 (2), 253-276, 2014.
[9] M. Özen and M. Güzeltepe, Codes over quaternion integers, Eur. J. Pure Appl. Math. 3 (4), 670-677, 2010.
[10] M. Özen and M. Güzeltepe, Cyclic codes over some finite quaternion integer rings, J. Franklin Inst. 348 (7), 1312-1317, 2011.
[11] D. Rohweder and S. Stern, Fischer, R.F.H., Shavgulidze, S., Freudenberger, J.: Four- Dimensional Hurwitz Signal Constellations, Set Partitioning, Detection, and Multi- level Coding, in: IEEE Transactions on Communications, 69 (8), 5079-5090, 2021.
[12] T. Shah and S.S. Rasool, On codes over quaternion integers, Appl. Algebra Engrg. Comm. Comput. 24 (6), 477-496, 2013.