Français
Projective Reed Muller codes revisited
Multi angle
Authors : Ghorpade, Sudhir (Author of the conference)
CIRM (Publisher )
14G15 - Finite ground fields
94B05 - Linear codes, general
Additional resources :
https://www.cirm-math.fr/RepOrga/2804/Slides/ALCOCRYPT-Ghorpade-2023.pdf
CIRM (Publisher )
Loading the player...
|
Abstract :
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.
Keywords : linear codes; Reed-Muller codes; minimum distance
MSC 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.
Keywords : linear codes; Reed-Muller codes; minimum distance
14G15 - Finite ground fields
94B05 - Linear codes, general
Additional resources :
https://www.cirm-math.fr/RepOrga/2804/Slides/ALCOCRYPT-Ghorpade-2023.pdf
|
Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 08/03/2023 Conference Date : 23/02/2023 Subseries : Research talks arXiv category : Information Theory ; Algebraic Geometry Mathematical Area(s) : Algebra ; Combinatorics Format : MP4 (.mp4) - HD Video Time : 00:34:34 Targeted Audience : Researchers ; Graduate Students ; Doctoral Students, Post-Doctoral Students Download : https://videos.cirm-math.fr/2023-02-23_Ghorpade.mp4 
|
Information on the Event
Event Title : ALCOCRYPTEvent Organizers : Bonnecaze, Alexis ; Mesnager, Sihem ; Solé, Patrick Dates : 20/02/2023 - 24/02/2023 Event Year : 2023 Event URL : https://conferences.cirm-math.fr/2804.html
Citation Data
DOI : 10.24350/CIRM.V.20006003Cite this video as: 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 |
See Also
- [Multi angle] Divisible codes / Author of the conference Kurz, Sascha.
- [Multi angle] Almost there: capacity of private information retrieval from coded and colluding servers / Author of the conference Hollanti, Camilla.
- [Multi angle] On APN and AB power functions / Author of the conference Budaghyan, Lilya.
Bibliography
- 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
Imagette Video
