# American Institute of Mathematical Sciences

2008, 2(1): 35-54. doi: 10.3934/amc.2008.2.35

## Duality theory for space-time codes over finite fields

 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  July 2007 Revised  January 2008 Published  January 2008

We further the study of the duality theory of linear space-time codes over finite fields by showing that the only finite linear ''omnibus'' codes (defined herein) with a duality theory are the column distance codes and the rank codes. We introduce weight enumerators for both these codes and show that they have MacWilliams-type functional equations relating them to the weight enumerators of their duals. We also show that the complete weight enumerator for finite linear sum-of-ranks codes satisfies such a functional equation. We produce an analogue of Gleason's Theorem for formally self-dual linear finite rank codes. Finally, we relate the duality matrices of $n\times n$ linear finite rank codes and length $n$ vector codes under the Hamming metric.
Citation: David Grant, Mahesh K. Varanasi. Duality theory for space-time codes over finite fields. Advances in Mathematics of Communications, 2008, 2 (1) : 35-54. doi: 10.3934/amc.2008.2.35
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