arXiv subjects
Publications and source records attributed to Rosemary Miguel Pires.
Code loops are Moufang loops constructed from doubly even binary codes. Then, given a code loop $L$, we ask which doubly even binary code $V$ produces $L$. In this sense, $V$ is called a representation of $L$. In this article we define and show how to determine all minimal and reduced representations of nonassociative code loops of rank $3$ and $4$.
For any code loop $L$, we prove that the half-automorphism group of $L$ is the product of the automorphism group of $L$ by an elementary abelian $2-$group consisting of all half-automorphisms that acts as the identity on a fixed basis. Also, we prove that elementary mappings only can be a half-automorphism on code loops of rank at most $3$.
We generalize the global construction of code loops introduced by Nagy, which is based on the connection between Moufang loops and groups with triality. This follows from the construction of a nilpotent group $G_n$ of class 3 with triality and $2n$ generators, based on embeddings of $G_n$ into direct products of copies of $G_3$. In the finite case, where $G_n$ is a group such that $|G_n| = 2^{4n+m}$ with $n \ge 3$ and $m = 3 {n \choose 2} + 2 {n \choose 3}$, we prove that the corresponding Moufang loop is the free loop $F_n$ with $n$ generators in the variety generated by code loops. The result depends on a construction similar to that of $G_n$, namely, embedding $F_n$ into direct products of copies of $F_3$, the free code loop associated with $G_3$.
Code loops are Moufang loops constructed from doubly even binary codes. Then, given a code loop L, we ask which doubly even binary code V produces L. In this sense, V is called a representation of L. In this article we define and determine all minimal and reduced representations of nonassociative code loops of rank 3 and 4.