We establish the link between correlation-tmmune functions and orthogonal arrays. We give a recursive definition of any correlation-immune function of maximal degree. We describe the set of quadratic balanced correlation-immune functions of maximal order. Some constructions are then deduced.

Abstract. We study [2 m −1, 2m]-binary linear codes whose weights lie between w 0 and 2 m −w 0 , where w 0 takes the highest possible value. Primitive cyclic codes with two zeros whose dual satisfies this property actually correspond to almost bent power functions and to pairs of maximum-length sequences with preferred crosscorrelation. We prove that, for odd m, these codes are completely characterized by their dual distance and by their weight divisibility. Using McEliece's theorem we give some general results on the weight divisibility of duals of cyclic codes with two zeros; specifically, we exhibit some infinite families of pairs of maximum-length sequences which are not preferred. [24] then reduces the determination of cyclic codes with two zeros whose dual is optimal to a purely combinatorial problem. It especially provides a very fast algorithm for finding such optimal cyclic codes, even for large lengths. These results also enables us to exhibit some infinite families of cyclic codes with two zeros whose dual is not optimal.This result widely applies in several areas of telecommunications: binary cyclic codes with two zeros whose dual is optimal in the previous sense are especially related to highly nonlinear power functions on finite fields; they also correspond to pairs of maximum-length sequences with preferred crosscorrelation. A function f from F 2 m into F 2 m is said to achieve the highest possible nonlinearity if any nonzero linear combination of its Boolean components is as far as possible from the set of Boolean affine functions with m variables. When m is odd, the highest possible value for the nonlinearity of a function over F 2 m is known and the functions achieving this bound are called almost bent (AB). These functions play a major role in cryptography; in particular, their use in the S-boxes of a Feistel cipher ensures the best resistance to

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