Higher genus universally decodable matrices (UDMG)

Steve Limburg; David Grant; Mahesh K. Varanasi

doi:10.3934/amc.2014.8.257

Article Contents

2014, Volume 8, Issue 3: 257-270. Doi: 10.3934/amc.2014.8.257

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Higher genus universally decodable matrices (UDMG)

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

Received: November 2012

Published online: August 2014

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Abstract

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:

Universally decodable matrices,
algebraic geometric codes.

Mathematics Subject Classification: Primary: 94B60, 94B05, 11T71.

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References

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