Higher genus universally decodable matrices (UDMG)

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August 2014, 8(3): 257-270. doi: 10.3934/amc.2014.8.257

Higher genus universally decodable matrices (UDMG)

Steve Limburg ^1, , David Grant ^1, and Mahesh K. Varanasi ^2,

1.	Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395, United States
2.	Department of Electrical and Computer Engineering, University of Colorado at Boulder, Boulder, CO 80309-0425, United States

Received November 2012 Published August 2014

Abstract
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We introduce the notion of Universally Decodable Matrices of Genus $g$ (UDMG), which for $g=0$ reduces to the notion of Universally Decodable Matrices (UDM) introduced in [8]. Fix positive $K,N,L$. A UDMG is a set $\{M_i|1\leq i\leq L\}$ of matrices of size $K \times N$ over a finite field such that the rows of any matrix of $K+g$ columns formed from the initial segments of the $M_i$ are linearly independent. We show that UDMG can be used to build approximately universal codes. We then provide a dictionary between UDMG and linear codes under the $m$-metric, which quickly provides constructions of UDMG and places bounds on the size of UDMG.

Keywords: algebraic geometric codes., Universally decodable matrices.

Mathematics Subject Classification: Primary: 94B60, 94B05, 11T7.

Citation: Steve Limburg, David Grant, Mahesh K. Varanasi. Higher genus universally decodable matrices (UDMG). Advances in Mathematics of Communications, 2014, 8 (3) : 257-270. doi: 10.3934/amc.2014.8.257

References:

[1]	S. T. Dougherty and M. M Skriganov, acWilliams duality and the Rosenbloom-Tsfasman metric, Mosc. Math. J., 2 (2002), 81-97.
[2]	A. Faldum and W. Willems, Codes of small defect, Des. Codes Crypt., 10 (1997), 341-350. doi: 10.1023/A:1008247720662.
[3]	A. Ganesan and N. Boston, Universally decodable matrices,, in Proc. 43rd Allerton Conf. Commun. Control Computing, (): 28.
[4]	A. Ganesan and P. O. Vontobel, On the existence of universally decodable matrices, IEEE Trans. Inform. Theory, 53 (2007), 2572-2575. doi: 10.1109/TIT.2007.899482.
[5]	R. Nielsen, A class of Sudan-decodable codes, IEEE Trans. Inform. Theory, 46 (2002), 1564-1572. doi: 10.1109/18.850696.
[6]	M. Y. Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Probl. Peredachi Inform., 33 (1997), 55-63.
[7]	J. H. Silverman, The Arithmetic of Elliptic Curves, Springer Verlag, 2009. doi: 10.1007/978-0-387-09494-6.
[8]	S. Tavildar and P. Viswanath, Approximately universal codes over slow-fading channels, IEEE Trans. Inform. Theory, 52 (2006), 3233-3258. doi: 10.1109/TIT.2006.876226.
[9]	D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. doi: 10.1017/CBO9780511807213.
[10]	P. O. Vontobel and A. Ganesan, On universally decodable matrices for space-time coding, Des. Codes Crypt., 41 (2006), 325-342. doi: 10.1007/s10623-006-9019-4.
[11]	J. L. Walker, Codes and Curves, Amer. Math. Soc., 2000.

show all references

References:

[1]	S. T. Dougherty and M. M Skriganov, acWilliams duality and the Rosenbloom-Tsfasman metric, Mosc. Math. J., 2 (2002), 81-97.
[2]	A. Faldum and W. Willems, Codes of small defect, Des. Codes Crypt., 10 (1997), 341-350. doi: 10.1023/A:1008247720662.
[3]	A. Ganesan and N. Boston, Universally decodable matrices,, in Proc. 43rd Allerton Conf. Commun. Control Computing, (): 28.
[4]	A. Ganesan and P. O. Vontobel, On the existence of universally decodable matrices, IEEE Trans. Inform. Theory, 53 (2007), 2572-2575. doi: 10.1109/TIT.2007.899482.
[5]	R. Nielsen, A class of Sudan-decodable codes, IEEE Trans. Inform. Theory, 46 (2002), 1564-1572. doi: 10.1109/18.850696.
[6]	M. Y. Rosenbloom and M. A. Tsfasman, Codes for the m-metric, Probl. Peredachi Inform., 33 (1997), 55-63.
[7]	J. H. Silverman, The Arithmetic of Elliptic Curves, Springer Verlag, 2009. doi: 10.1007/978-0-387-09494-6.
[8]	S. Tavildar and P. Viswanath, Approximately universal codes over slow-fading channels, IEEE Trans. Inform. Theory, 52 (2006), 3233-3258. doi: 10.1109/TIT.2006.876226.
[9]	D. Tse and P. Viswanath, Fundamentals of Wireless Communication, Cambridge University Press, 2005. doi: 10.1017/CBO9780511807213.
[10]	P. O. Vontobel and A. Ganesan, On universally decodable matrices for space-time coding, Des. Codes Crypt., 41 (2006), 325-342. doi: 10.1007/s10623-006-9019-4.
[11]	J. L. Walker, Codes and Curves, Amer. Math. Soc., 2000.

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