Combinatorial polarization, code loops, and codes of high level

Abstract

We first find the combinatorial degree of any map $f:V\to F$ where $F$ is a finite field and $V$ is a finite-dimensional vector space over $F$. We then simplify and generalize a certain construction due to Chein and Goodaire that was used in characterizing code loops as finite Moufang loops that posses at most two squares. The construction yields binary codes of high divisibility level with prescribed Hamming weights of intersections of codewords.