Group structure on projective spaces and cyclic codes over finite fields

Abstract : We study the geometrical properties of the subgroups of the multiplicative group of a finite extension of a finite field endowed with its vector space structure, and we show that in some cases the associated projective space has a natural groupe structure. We construct some cyclic codes related to Reed-Muller codes by evaluating polynomials on these subgroups. The geometrical properties of these groups give a fairly simple description of these codes of the Reed-Muller kind.
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Submitted on : Friday, January 4, 2013 - 6:51:19 PM
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Gilles Lachaud, Isabelle Lucien, Dany-Jack Mercier, Robert Rolland. Group structure on projective spaces and cyclic codes over finite fields. Finite Fields and Their Applications, Elsevier, 2000, 6 (2), pp.119-129. ⟨10.1006/ffta.1999.0268⟩. ⟨hal-00770256⟩

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