Symmetries of weight enumerators and applications to Reed-Muller codes

Borello, Martino; Mila, Olivier Bernard (2019). Symmetries of weight enumerators and applications to Reed-Muller codes. Advances in mathematics of communications, 13(2), pp. 313-328. American Institute of Mathematical Sciences 10.3934/amc.2019021

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Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics

UniBE Contributor:

Mila, Olivier Bernard

Subjects:

500 Science > 510 Mathematics

ISSN:

1930-5346

Publisher:

American Institute of Mathematical Sciences

Language:

English

Submitter:

Michel Arthur Bik

Date Deposited:

06 Aug 2019 14:17

Last Modified:

25 Oct 2019 02:44

Publisher DOI:

10.3934/amc.2019021

ArXiv ID:

1707.00575v2

BORIS DOI:

10.7892/boris.132261

URI:

https://boris.unibe.ch/id/eprint/132261

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