H-codes and Derivations


Let m be a positive integer. In this text, a Boolean function is a map from  into the field  . The affine invariants are tools to progress in the knowledge of the Boolean functions. An affine invariant j, is a map such that  , for all Boolean functions f and all affine transformations  of  . For example, the weight, the degree and the non-linearity are affine invariants. In this paper, we study a particular affine invariant : the height. In his thesis, J Dillon defines the derivation of a Boolean function f in the direction of a subspace S as  . The height of f, denoted by  , is equal to the minimal dimension of a subspace S such that  is the null function. The expession  is nothing but the convolutional product of f and the indicating function of S; It is also their product in the group algebra  . The height of f gives an information about the minimal distance of the annihilator of f. Following a definition of P. Camion, an H-code is a principal ideal of dimension  . An H-code is self-dual, and equal to its annihilator. In her article Self-dual codes which are principal ideals of the group algebra, P. Charpin proves that the family of H-codes is not a class of good codes, showing that the maximal value of the height of a function is  . Here, we give some values and new bounds on the strange function

P. Langevin, Octobre 1998