@MISC{Y_derivedforms, author = {Petr Vojt Ěchovsk Y}, title = {DERIVED FORMS AND BINARY LINEAR CODES}, year = {} }

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Abstract

Abstract. Derived forms defined by M. Aschbacher in [1] are closely related to combinatorial polarization introduced by H. N. Ward in [6]. A binary linear code is said to be of (divisibility) level r, if r is the biggest integer such that 2 r divides the weight of each codeword. In this paper, we study the relation between functions of combinatorial degreee r+1 and binary linear codes of level r. We associate a code of level at least r with every function of combinatorial degree r + 1 (as a generalization of the case r = 2 considered by O. Chein and E. Goodaire), and vice versa. Several examples are provided. 1. Derived Forms Throughout this paper let V denote a vector space of dimension m over two element field F = {0, 1}. Since V is a subspace of a ring F n = (F n, +, ∗), where multiplication ∗ is defined component-wise, it is reasonable to consider the product u ∗ v of two vectors u, v ∈ V. Instead of sets we will use labeled sets, i.e. ordered collections of elements with possible repetitions. Unlike in [6] (where the formal definition of labeled sets is given) we will not introduce a special notation for labeled sets but we will always keep in mind that we do not deal with ordinary sets. For a labeled set of vectors I = {v1,..., vn} define � I = v1 + · · · + vn, and � I = v1 ∗ · · · ∗ vn. Let us agree on � ∅ = � ∅ = 0. Also, besides its usual set theoretical meaning, let |v | denote the number of non-zero coordinates of v ∈ V, the weight of v. In 1979, H. N. Ward defined the combinatorial polarization df of a mapping f: A − → B between abelian groups A, B by df(S) = �