Littlewood polynomials with high order zeros

Let N∗(m) be the minimal length of a polynomial with ±1 coefficients divisible by (x − 1)m. Byrnes noted that N∗(m) ≤ 2m for each m, and asked whether in fact N∗(m) = 2m. Boyd showed that N∗(m) = 2m for all m ≤ 5, but N∗(6) = 48. He further showed that N∗(7) = 96, and that N∗(8) is one of the 5 numbers 96, 144, 160, 176, or 192. Here we prove that N∗(8) = 144. Similarly, let m∗(N) be the maximal power of (x− 1) dividing some polynomial of degree N − 1 with ±1 coefficients. Boyd was able to find m∗(N) for N < 88. In this paper we determine m∗(N) for N < 168.