Section: New Results
Rank metric codes over infinite fields
Rank metric and Gabidulin codes over the rationals promise
interesting applications to space-time coding. We have constructed
optimal codes, similar to Gabidulin codes, in the case of infinite
fields. We use algebraic extensions, and we have determined the
condition on the considered extension to enable this construction.
For example: we can design codes with complex coefficients, using
number fields and Galois automorphisms.
Then, in the rank metric setting, codewords can be seen as matrices.
In this setting, a channel introduces errors (a matrix of small rank
We also have used this framework to build rank-metric codes over the field of rational functions, using algebraic function fields with cyclic Galois group (Kummer and Artin extensions). These codes can be seen as a generator of infinitely many convolutional codes [25] .