On packings of $(n,k)$-products

An $(n, k)$-product (or simply a product), $n\ge 2k$, is the product of $k$ binomials on the set of $n$ variables; the variables in the product are not repeated. The decomposition of a product is the set of $2^k$ monomials of length $k$ appearing after expanding the brackets in this product. The sum of some products is called a packing if after the decomposition of all products in this sum every monomial appears at most once. The length of the sum of products is the number of products in this sum. A packing is called perfect if every possible monomial of length $k$ appears exactly once. The problem of packings is motivated by the construction of Boolean functions with cryptographically important properties. In the paper we give recursive constructions of packings of products (including perfect ones) and the corresponding recurrence bounds on their length. We give necessary conditions on the parameters $n$ and $k$ for the existence of a perfect packing of $(n, k)$-products. We give the complete solution of the problem of the existence of perfect packings of $(n,k)$-products for $k\le 3$. We find the exact value for the maximal length of a packing of $(n, 2)$-products for any $n$.