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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

Jérôme Ducoat 1, and  Frédérique Oggier 1,

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.
Keywords: division algebras, space-time codes., Constacyclic codes, skew polynomial rings.
Mathematics Subject Classification: Primary: 11S45; Secondary: 11T71, 94B4.
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
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J.-C. Belfiore and F. Oggier, An error probability approach to MIMO wiretap channels, IEEE Trans. Commun., 61 (2013), 3396-3403. Google Scholar

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G. Berhuy and F. Oggier, An Introduction to Central Simple Algebras and Their Applications to Wireless Communication, AMS, 2013. doi: 10.1090/surv/191.  Google Scholar

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D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.  Google Scholar

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D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

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D. Boucher, F. Ulmer and P. Solé, Skew constacyclic codes over Galois rings, Adv. Math. Commun., 2 (2008), 273-292. doi: 10.3934/amc.2008.2.273.  Google Scholar

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J. H. Conway and N. J. A Sloane, Sphere Packings, Lattices and Groups,, Springer., ().   Google Scholar

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J. Ducoat and F. Oggier, Lattice encoding of cyclic codes from skew-polynomial rings, in Coding Theory and Applications, Springer, 2015, 161-167. Google Scholar

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W. Ebeling, Lattices and Codes, A Course Partially Based on Lectures by Friedrich Hirzebruch, Springer, 2013. doi: 10.1007/978-3-658-00360-9.  Google Scholar

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G. D. Forney, Coset Codes Part I: Introduction and geometrical classification, IEEE Trans. Inf. Theory, 34 (1988), 1123-1151. doi: 10.1109/18.21245.  Google Scholar

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F. Oggier and J.-C. Belfiore, Enabling multiplication in lattice codes via construction A, in IEEE Int. Workshop Inf. Theory, 2013, 1-5. Google Scholar

[14]

F. Oggier and B. A. Sethuraman, Quotients of orders in cyclic algebras and space-time codes, Adv. Math. Commun., 7 (2013), 441-461. doi: 10.3934/amc.2013.7.441.  Google Scholar

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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), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

show all references

References:
[1]

M. Artin, Noncommutative Rings,, 1999., ().   Google Scholar

[2]

C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A, 78 (1997), 92-119. doi: 10.1006/jcta.1996.2763.  Google Scholar

[3]

J.-C. Belfiore and F. Oggier, An error probability approach to MIMO wiretap channels, IEEE Trans. Commun., 61 (2013), 3396-3403. Google Scholar

[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.  Google Scholar

[5]

A. Bonnecaze, P. Solé and A. R. Calderbank, Quaternary qudratic residue codes and unimodular lattices, IEEE Trans. Inf. Theory, 41 (1995), 366-377. doi: 10.1109/18.370138.  Google Scholar

[6]

D. Boucher, W. Geiselmann and F. Ulmer, Skew cyclic codes, Appl. Algebra Engin. Commun. Comput., 18 (2007), 379-389. doi: 10.1007/s00200-007-0043-z.  Google Scholar

[7]

D. Boucher and F. Ulmer, Coding with skew polynomial rings, J. Symb. Comput., 44 (2009), 1644-1656. doi: 10.1016/j.jsc.2007.11.008.  Google Scholar

[8]

D. Boucher, F. Ulmer and P. Solé, Skew constacyclic codes over Galois rings, 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]

J. Ducoat and F. Oggier, Lattice encoding of cyclic codes from skew-polynomial rings, in Coding Theory and Applications, Springer, 2015, 161-167. Google Scholar

[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.  Google Scholar

[12]

G. D. Forney, Coset Codes Part I: Introduction and geometrical classification, IEEE Trans. Inf. Theory, 34 (1988), 1123-1151. doi: 10.1109/18.21245.  Google Scholar

[13]

F. Oggier and J.-C. Belfiore, Enabling multiplication in lattice codes via construction A, in IEEE Int. Workshop Inf. Theory, 2013, 1-5. Google Scholar

[14]

F. Oggier and B. A. Sethuraman, Quotients of orders in cyclic algebras and space-time codes, Adv. Math. Commun., 7 (2013), 441-461. doi: 10.3934/amc.2013.7.441.  Google Scholar

[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), 2596-2616. doi: 10.1109/TIT.2003.817831.  Google Scholar

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