# American Institute of Mathematical Sciences

February  2016, 10(1): 79-94. doi: 10.3934/amc.2016.10.79

## On skew polynomial codes and lattices from quotients of cyclic division algebras

 1 Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

Received  December 2014 Revised  August 2015 Published  March 2016

We propose a variation of Construction A of lattices from linear codes defined using the quotient $\Lambda/\mathfrak{p}\Lambda$ of some order $\Lambda$ inside a cyclic division $F$-algebra, for $\mathfrak{p}$ a prime ideal of a number field $F$. To obtain codes over this quotient, we first give an isomorphism between $\Lambda/\mathfrak{p}\Lambda$ and a ring of skew polynomials. We then discuss definitions and basic properties of skew polynomial codes, which are needed for Construction A, but also explore further properties of the dual of such codes. We conclude by providing an application to space-time coding, which is the original motivation to consider cyclic division $F$-algebras as a starting point for this variation of Construction A.
Citation: Jérôme Ducoat, Frédérique Oggier. On skew polynomial codes and lattices from quotients of cyclic division algebras. Advances in Mathematics of Communications, 2016, 10 (1) : 79-94. doi: 10.3934/amc.2016.10.79
##### References:
 [1] M. Artin, Noncommutative Rings,, 1999., (). [2] C. Bachoc, Applications of coding theory to the construction of modular lattices,, J. Combin. Theory Ser. A, 78 (1997), 92. doi: 10.1006/jcta.1996.2763. [3] J.-C. Belfiore and F. Oggier, An error probability approach to MIMO wiretap channels,, IEEE Trans. Commun., 61 (2013), 3396. [4] G. Berhuy and F. Oggier, An Introduction to Central Simple Algebras and Their Applications to Wireless Communication,, AMS, (2013). doi: 10.1090/surv/191. [5] A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary qudratic residue codes and unimodular lattices,, IEEE Trans. Inf. Theory, 41 (1995), 366. doi: 10.1109/18.370138. [6] D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes,, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. [7] D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symb. Comput., 44 (2009), 1644. doi: 10.1016/j.jsc.2007.11.008. [8] D. Boucher, F. Ulmer and P. Solé, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273. doi: 10.3934/amc.2008.2.273. [9] J. H. Conway and N. J. A Sloane, Sphere Packings, Lattices and Groups,, Springer., (). [10] J. Ducoat and F. Oggier, Lattice encoding of cyclic codes from skew-polynomial rings,, in Coding Theory and Applications, (2015), 161. [11] W. Ebeling, Lattices and Codes, A Course Partially Based on Lectures by Friedrich Hirzebruch,, Springer, (2013). doi: 10.1007/978-3-658-00360-9. [12] G. D. Forney, Coset Codes Part I: Introduction and geometrical classification,, IEEE Trans. Inf. Theory, 34 (1988), 1123. doi: 10.1109/18.21245. [13] F. Oggier and J.-C. Belfiore, Enabling multiplication in lattice codes via construction A,, in IEEE Int. Workshop Inf. Theory, (2013), 1. [14] F. Oggier and B. A. Sethuraman, Quotients of orders in cyclic algebras and space-time codes,, Adv. Math. Commun., 7 (2013), 441. doi: 10.3934/amc.2013.7.441. [15] B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras,, IEEE Trans. Inf. Theory, 49 (2003), 2596. doi: 10.1109/TIT.2003.817831.

show all references

##### References:
 [1] M. Artin, Noncommutative Rings,, 1999., (). [2] C. Bachoc, Applications of coding theory to the construction of modular lattices,, J. Combin. Theory Ser. A, 78 (1997), 92. doi: 10.1006/jcta.1996.2763. [3] J.-C. Belfiore and F. Oggier, An error probability approach to MIMO wiretap channels,, IEEE Trans. Commun., 61 (2013), 3396. [4] G. Berhuy and F. Oggier, An Introduction to Central Simple Algebras and Their Applications to Wireless Communication,, AMS, (2013). doi: 10.1090/surv/191. [5] A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary qudratic residue codes and unimodular lattices,, IEEE Trans. Inf. Theory, 41 (1995), 366. doi: 10.1109/18.370138. [6] D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes,, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379. doi: 10.1007/s00200-007-0043-z. [7] D. Boucher and F. Ulmer, Coding with skew polynomial rings,, J. Symb. Comput., 44 (2009), 1644. doi: 10.1016/j.jsc.2007.11.008. [8] D. Boucher, F. Ulmer and P. Solé, Skew constacyclic codes over Galois rings,, Adv. Math. Commun., 2 (2008), 273. doi: 10.3934/amc.2008.2.273. [9] J. H. Conway and N. J. A Sloane, Sphere Packings, Lattices and Groups,, Springer., (). [10] J. Ducoat and F. Oggier, Lattice encoding of cyclic codes from skew-polynomial rings,, in Coding Theory and Applications, (2015), 161. [11] W. Ebeling, Lattices and Codes, A Course Partially Based on Lectures by Friedrich Hirzebruch,, Springer, (2013). doi: 10.1007/978-3-658-00360-9. [12] G. D. Forney, Coset Codes Part I: Introduction and geometrical classification,, IEEE Trans. Inf. Theory, 34 (1988), 1123. doi: 10.1109/18.21245. [13] F. Oggier and J.-C. Belfiore, Enabling multiplication in lattice codes via construction A,, in IEEE Int. Workshop Inf. Theory, (2013), 1. [14] F. Oggier and B. A. Sethuraman, Quotients of orders in cyclic algebras and space-time codes,, Adv. Math. Commun., 7 (2013), 441. doi: 10.3934/amc.2013.7.441. [15] B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras,, IEEE Trans. Inf. Theory, 49 (2003), 2596. doi: 10.1109/TIT.2003.817831.

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