473 research outputs found
Symmetries of weight enumerators and applications to Reed-Muller codes
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.Comment: 14 pages. Improved and extended version of arXiv:1511.00803. To
appear in Advances in Mathematics of Communication
Weight enumerators of Reed-Muller codes from cubic curves and their duals
Let be a finite field of characteristic not equal to or
. We compute the weight enumerators of some projective and affine
Reed-Muller codes of order over . These weight enumerators
answer enumerative questions about plane cubic curves. We apply the MacWilliams
theorem to give formulas for coefficients of the weight enumerator of the duals
of these codes. We see how traces of Hecke operators acting on spaces of cusp
forms for play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and
Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math.,
201
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