Let n be an odd prime and m > 1 be a positive integer. We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree 2 m and length 2 n + 1. Some examples are given to illustrate our results.

The weight, balancedness and nonlinearity are important properties of Boolean functions, but they can be difficult to determine in general. In this paper, we study how to compute them for two classes of functions where these problems are more tractable. In particular, we study functions of degree three and the so-called “splitting” functions. The latter are functions that can be written as the sum of two functions defined over disjoint sets of variables. We show how, for splitting functions, studying these properties reduces to the study of simpler functions. We provide then a procedure to compute the weight of a cubic Boolean function. We show computationally that, for a cubic Boolean function with limited number of terms, this procedure is on average significantly more efficient than some other methods.

A nonzero nonunit a of a ring R is called an irreducible element if, for some b, c ∈ R, a = bc implies that either b or c (not both) is a unit. We construct a bipartite graph in which the union of the set of irreducible elements and group of units is a vertex-set and an edge-set is the set of pairs between irreducible elements and their unit factors in the ring of integers modulo n. Many properties of this constructed bipartite graph are studied. We show that this bipartite graph contains components which are isomorphic. We also note that each component of this bipartite graph can be presented in some form which we call star form presentation. Some examples of graphs in star form presentation are provided for illustration purposes. Furthermore, we prove that the girth of this bipartite graph is 8. Most of the results in this paper are arrived at via group action.

Let n and be odd prime numbers such that = n and ( , 2 n ± 1) = 1. We produce an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree (2 ) m , with m ≥ 1 and length 2 n + 1.