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The ring in the title is the first non commutative ring to have been used asalphabet for block codes. The original motivation was the construction of somequaternionic modular lattices from codes. The new application is theconstruction of space time codes obtained by concatenation from the Goldencode. In this article, we derive structure theorems for cyclic codes over thatring, and use them to characterize the lengths where self dual cyclic codesexist. These codes in turn give rise to formally self dual quaternary codes.

Cyclic codes over $M_2(\F_2)$