A Note on the Games-Chan Algorithm

The Games-Chan algorithm finds the minimal period of a periodic binary sequence of period 2 n , in n iterations. We generalise this to periodic q -ary sequences (where q is a prime power) using generating functions and polynomials. We apply this to find the multiplicity of x − 1 in a q -ary polynomial f in log q deg( f ) iterations.

💡 Research Summary

The paper “A Note on the Games‑Chan Algorithm” revisits the classic Games‑Chan method for determining the minimal period of a binary sequence whose period is a power of two, and extends the technique to sequences over any finite field GF(q) where q is a prime power. The original binary algorithm works in n iterations for a period N = 2ⁿ by exploiting the generating function G(s)(x) = Σ s_i x^i and the relationship between the minimal polynomial µ(s) = (x^N‑1)/gcd(s, x^N‑1) and the minimal period mp(s) = deg µ(s). The authors observe that µ(s) can be expressed via the greatest common divisor of the sequence’s polynomial representation, and that mp(s) = N – |gcd(s, x^N‑1)|.

To generalize, the authors introduce the notion of “q‑folding”. For a sequence of length N = q·N′, the sequence is partitioned into q blocks s(0), …, s(q‑1), each of length N′. They define a folded sequence s• = s(0) + … + s(q‑2) + x^{‑(q‑1)N′} s(q‑1), which has length N′. Theorem 1.2 shows that the generating function of the original sequence can be written as a linear combination of the block generating functions with appropriate shifts, and that the minimal period obeys a simple recursion:

• If s• = 0 then mp(s) = max{mp(s(0)), …, mp(s(q‑2))}.
• If s• ≠ 0 then mp(s) = mp(s•) + (q‑1)N′. Thus each recursion reduces the length by a factor of q, leading to a logarithmic‑in‑q number of steps. The proof relies on properties of gcd and lcm in the polynomial ring GF(q)