# American Institute of Mathematical Sciences

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

## Higher genus universally decodable matrices (UDMG)

 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

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.
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
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