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., ().   Google Scholar [2] J. Combin. Theory Ser. A, 78 (1997), 92-119. doi: 10.1006/jcta.1996.2763.  Google Scholar [3] IEEE Trans. Commun., 61 (2013), 3396-3403. Google Scholar [4] AMS, 2013. doi: 10.1090/surv/191.  Google Scholar [5] IEEE Trans. Inf. Theory, 41 (1995), 366-377. doi: 10.1109/18.370138.  Google Scholar [6] Appl. Algebra Engin. Commun. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.  Google Scholar [7] J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar [8] Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.  Google Scholar [9] J. H. Conway and N. J. A Sloane, Sphere Packings, Lattices and Groups,, Springer., ().   Google Scholar [10] in Coding Theory and Applications, Springer, 2015, 161-167. Google Scholar [11] Springer, 2013. doi: 10.1007/978-3-658-00360-9.  Google Scholar [12] IEEE Trans. Inf. Theory, 34 (1988), 1123-1151. doi: 10.1109/18.21245.  Google Scholar [13] in IEEE Int. Workshop Inf. Theory, 2013, 1-5. Google Scholar [14] Adv. Math. Commun., 7 (2013), 441-461. doi: 10.3934/amc.2013.7.441.  Google Scholar [15] IEEE Trans. Inf. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

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References:
 [1] M. Artin, Noncommutative Rings,, 1999., ().   Google Scholar [2] J. Combin. Theory Ser. A, 78 (1997), 92-119. doi: 10.1006/jcta.1996.2763.  Google Scholar [3] IEEE Trans. Commun., 61 (2013), 3396-3403. Google Scholar [4] AMS, 2013. doi: 10.1090/surv/191.  Google Scholar [5] IEEE Trans. Inf. Theory, 41 (1995), 366-377. doi: 10.1109/18.370138.  Google Scholar [6] Appl. Algebra Engin. Commun. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.  Google Scholar [7] J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar [8] Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.  Google Scholar [9] J. H. Conway and N. J. A Sloane, Sphere Packings, Lattices and Groups,, Springer., ().   Google Scholar [10] in Coding Theory and Applications, Springer, 2015, 161-167. Google Scholar [11] Springer, 2013. doi: 10.1007/978-3-658-00360-9.  Google Scholar [12] IEEE Trans. Inf. Theory, 34 (1988), 1123-1151. doi: 10.1109/18.21245.  Google Scholar [13] in IEEE Int. Workshop Inf. Theory, 2013, 1-5. Google Scholar [14] Adv. Math. Commun., 7 (2013), 441-461. doi: 10.3934/amc.2013.7.441.  Google Scholar [15] IEEE Trans. Inf. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar
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