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Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou [28] in even characteristic. In 2016, L. Qu [23] proposed a new approach to constructing quadratic planar functions over F 2 n . Very recently, D. Bartoli and M. Timpanella [4] characterized the condition on coefficients a , b such that the function f a , b (x) = a x 2 2 m + 1 + b x 2 m + 1 ∈ F 2 3 m [ x ] is a planar function over F 2 3 m by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in [23] , we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over F q k , where q = 2 m with m sufficiently large (see Theorem 1.1). The first and last classes of them are over F q 2 and F q 4 respectively, while the other two classes are over F q 3 . One class over F q 3 is an extension of f a , b (x) investigated in [4] , while our proofs seem to be much simpler. In addition, although the planar binomial over F q 2 of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in [23]. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a finitely presented group, and let p be a prime. Then G is ‘large’ (respectively, ‘p-large’) if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a variety of new methods for detecting whether G is large or p-large. These relate to the group''s profinite and pro-p completions, to its first -Betti number and to the existence of certain finite index subgroups with ‘rapid descent’. The paper draws on new topological and geometric techniques, together with a result on error-correcting codes. [Copyright &y& Elsevier]