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LFANT - 2018

Section: New Results

Error-correcting codes based on non-commutative algebras

Participant : Aurel Page.

In [36], C. Maire and A. Page revisit a construction due to Lenstra and Guruswami by generalising them to unit groups of division algebras.

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, Maire and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. Maire and Page prove that the noncommutative unit group construction yields asymptotically good families of codes for division algebras of any degree, and estimate the smallest possible size of the alphabet in terms of the degree of the algebra.