On almost balanced Boolean functions

A Boolean function is called correlation-immune of degree $n-m$ if it takes the value $1$ the same number of times for each $m$-dimensional face of the hypercube. Balanced correlation-immune function is called resilient. The almost balanced (or almost resilient) Boolean function is defined as a function taking values $1$ in a half or in a half plus or minus one of vertices in each face. Here, some constructions of almost balanced functions are proposed, some properties and a low bound for the number of these functions are established.