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Ore polynomials and applications to coding theory

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Ore polynomials and applications to coding theory
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7
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CC Attribution - NonCommercial - NoDerivatives 2.0 Generic:
You are free to use, copy, distribute and transmit the work or content in unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
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In the 1930’s, in the course of developing non-commutative algebra, Ore introduced a twisted version of polynomials in which the scalars do not commute with the variable. About fifty years later, Delsarte, Rothand Gabidulin realized (independently) that Ore polynomials could be used to define codes—nowadays calledGabidulin codes—exhibiting good properties with respect to the rank distance. More recently, Gabidulincodes have received much attention because of many promising applications to network coding, distributedstorage and cryptography. The first part of my talk will be devoted to review the classical construction of Gabidulin codes and presenta recent extension due to Martinez-Penas and Boucher (independently), offering similar performances butallowing for transmitting much longer messages in one shot. I will then revisit Martinez-Penas’ and Boucher’sconstructions and give to them a geometric flavour. Based on this, I will derive a geometric description ofduals of these codes and finally speculate on the existence of more general geometric Gabidulin codes.
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