Littlewood Polynomials, Spectral-Null Codes, and Equipowerful Partitions

Let $[n]$ denote $\{0,1, ... , n-1\}$. A polynomial $f(x) = \sum a_i x^i$ is a Littlewood polynomial (LP) of length $n$ if the $a_i$ are $\pm 1$ for $i \in [n]$, and $a_i = 0$ for $i \ge n$. Such an LP is said to have order $m$ if it is divisible by $(x-1)^m$. The problem of finding the set $L_m$ of lengths of LPs of order $m$ is equivalent to finding the lengths of spectral-null codes of order $m$, and to finding $n$ such that $[n]$ admits a partition into two subsets whose first $m$ moments are equal. Extending the techniques and results of Boyd and others, we completely determine $L_7$ and $L_8$ and prove that 192 is the smallest element of $L_9$. Our primary tools are the use of carefully targeted searches using integer linear programming (both to find LPs and to disprove their existence for specific $n$ and $m$), and an unexpected new concept (that arose out of observed symmetry properties of LPs) that we call "regenerative pairs," which produce infinite arithmetic progressions in $L_m$. We prove that for $m \le$ 8, whenever there is an LP of length $n$ and order $m$, there is one of length $n$ and order $m$ that is symmetric (resp.~antisymmetric) if m is even (resp.~odd).