The classification of antipodal two-weight linear codes

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By a classical result of Bonisoli, the equidistant linear codes over GF(q) are, up to monomial equivalence, just the replications of some q-ary simplex code, possibly with added 0-coordinates. We prove an analogous result for the antipodal two-weight linear codes over GF(q) (that is, one of the two weights is the length of the code): up to monomial equivalence, any such code is ether a replication of a first order generalized Reed–Muller code, or a replication of a 3-dimensional projective code associated with some non-trivial maximal arc in the classical projective plane PG(2,2s), s≥1.

An equivalent geometric formulation of this result reads as follows: a multiset S of points spanning Π=PG(m,q), m≥2, for which every hyperplane intersecting S does so in a constant number of points (counted with multiplicity), is either a multiple of a maximal arc in Π (for m=2) or a multiple of the complement of some hyperplane of Π.

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© 2017 Elsevier Inc. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.ffa.2017.12.010

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Finite Fields and Their Applications