Littlewood polynomials, spectral-null codes, and equipowerful partitions

Let [n] denote {0,1, …,n — 1}. A polynomial f(x) = ∑ a_iXⁱ is a Littlewood polynomial (LP) of length n if the a_i are ±1 for iϵ [n], and a_i = 0 for i > n. An LP has order m if it is divisible by (x — l)^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 initiated by Boyd, we completely determine L7 and Ls, and prove that min L₉ = 192 and min L₁₀ = 240. Our primary tools are (a) symmetry, and (b) the use of carefully targeted searches using integer linear programming both to find LPs and to disprove their existence for specific n and m. Symmetry plays an unexpected role and leads to the concept of regenerative pairs, which produce infinite arithmetic progressions in L_m. We prove for m ≤ 8 (and conjecture later for all m) that 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).