English
Projective Reed Muller codes revisited
Multi angle
Auteurs : Ghorpade, Sudhir (Auteur de la conférence)
CIRM (Editeur )
14G15 - Finite ground fields
94B05 - Linear codes, general
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2804/Slides/ALCOCRYPT-Ghorpade-2023.pdf
CIRM (Editeur )
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Résumé :
Projective Reed Muller Codes constitute an interesting class of linear codes, which was introduced by Gilles Lachaud in 1988. Questions about their minimum distance are intimately related to the question about the maximum possible number of F-rational points in the m-dimensional projective space on a hypersurface of degree d in m+1 variables with coefficients in a finite field F. Michael Tsfasman gave a conjectural formula for this maximum possible number of points on such hypersurfaces, and the conjecture was soon proved in the affirmative by Jean-Pierre Serre. In all these works, it is generally assumed that the degree d is at most q, where q is the number of elements in F. Anders Sørensen considered in 1991 more general projective Reed Muller codes where d can be larger than q. From a coding theoretical perspective, it is more natural to consider this larger class. Sørensen proposed a formula for the minimum distance in the general case, and also studied the duals of the projective Reed-Muller codes.
We shall revisit the work of Sorensen by pointing out some minor inaccuracies in his proof of the minimum distance. We then propose an alternative proof. Further, we address the question of obtaining a characterization of the minimum weight codewords of projective Reed Muller codes.
This is a joint work with Rati Ludhani.
Mots-Clés : linear codes; Reed-Muller codes; minimum distance
Codes MSC : We shall revisit the work of Sorensen by pointing out some minor inaccuracies in his proof of the minimum distance. We then propose an alternative proof. Further, we address the question of obtaining a characterization of the minimum weight codewords of projective Reed Muller codes.
This is a joint work with Rati Ludhani.
Mots-Clés : linear codes; Reed-Muller codes; minimum distance
14G15 - Finite ground fields
94B05 - Linear codes, general
Ressources complémentaires :
https://www.cirm-math.fr/RepOrga/2804/Slides/ALCOCRYPT-Ghorpade-2023.pdf
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Informations sur la Vidéo
Réalisateur : Hennenfent, GuillaumeLangue : Anglais Date de Publication : 08/03/2023 Date de Captation : 23/02/2023 Sous Collection : Research talks Catégorie arXiv : Information Theory ; Algebraic Geometry Domaine(s) : Algèbre ; Combinatoires Format : MP4 (.mp4) - HD Durée : 00:34:34 Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023-02-23_Ghorpade.mp4 
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Informations sur la Rencontre
Nom de la Rencontre : ALCOCRYPTOrganisateurs de la Rencontre : Bonnecaze, Alexis ; Mesnager, Sihem ; Solé, Patrick Dates : 20/02/2023 - 24/02/2023 Année de la rencontre : 2023 URL de la Rencontre : https://conferences.cirm-math.fr/2804.html
Données de citation
DOI : 10.24350/CIRM.V.20006003Citer cette vidéo: Ghorpade, Sudhir (2023). Projective Reed Muller codes revisited. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20006003 URI : http://dx.doi.org/10.24350/CIRM.V.20006003 |
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Bibliographie
- BEELEN, Peter, DATTA, Mrinmoy, et GHORPADE, Sudhir R. A combinatorial approach to the number of solutions of systems of homogeneous polynomial equations over finite fields. Moscow Mathematical Journal, 2022, vol. 22, no 4, p. 565-593. - http://dx.doi.org/10.17323/1609-4514-2022-22-4-565-593
- DATTA, Mrinmoy et GHORPADE, Sudhir R. Remarks on the Tsfasman-Boguslavsky Conjecture and higher weights of projective Reed-Muller codes. In : Arithmetic, geometry, cryptography and coding theory. 2017. p. 157-169. - https://www.ams.org/books/conm/686/13782
- DELSARTE, Philippe, GOETHALS, Jean-Marie, et MAC WILLIAMS, F. Jessie. On generalized reedmuller codes and their relatives. Information and control, 1970, vol. 16, no 5, p. 403-442. - https://doi.org/10.1016/S0019-9958(70)90214-7
- LACHAUD, Gilles. Projective reed-muller codes. In : Coding Theory and Applications: 2nd International Colloquium Cachan-Paris, France, November 24–26, 1986 Proceedings 2. Springer Berlin Heidelberg, 1988. p. 125-129. - http://dx.doi.org/10.1007/3-540-19368-5_13
- LACHAUD, Gilles. The parameters of projective Reed–Müller codes. Discrete Mathematics, 1990, vol. 81, no 2, p. 217-221. - https://doi.org/10.1016/0012-365X(90)90155-B
- SERRE, J. P. et TSFASMAN, Lettrea M. Journées Arithmétiques (Luminy, 1989). Astérisque, 1991, vol. 198, p. 149-156. - http://www.numdam.org/item/AST_1991__198-199-200__351_0/
- SORENSEN, Anders Bjaert. Projective Reed-Muller codes. IEEE Transactions on Information Theory, 1991, vol. 37, no 6, p. 1567-1576. - https://doi.org/10.1109/18.104317
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