Asymptotically-Tight Bounds on the Number of Cycles in Generalized de Bruijn-Good Graphs

Asymptotically-Tight Bounds on the Number of Cycles in Generalized de Bruijn-Good Graphs Ueli M. Maurer Institute for Signal and Information Processing Swiss Federal Institute of Technology CH-8092 Z urich, Switzerland Abstract. The number of cycles of length k that can be generated by q-ary n-stage feedback shift-registers is studied. This problem is equivalent to nding the number of cycles of length k in the natural generalization, from binary to q-ary digits, of the so-called de Bruijn-Good graphs [2, 7]. The number of cycles of length k in the q-ary graph G(q) n of order n is denoted by (q)(n; k). Known results about (2)(n; k) are summarized and extensive new numerical data is presented. Lower and upper bounds on (q)(n; k) are derived showing that, for large k, virtually all q-ary cycles of length k are contained in G(q) n for n > 2 logq k, but virtually none of these cycles is contained in G(q) n for n < 2 logq k 2 logq logq k. More precisely, if (q) k denotes the total number of q-ary length k cycles, then for any function f(k) that grows without bounds as k ! 1 (e.g. f(k) = logq logq logq k), the bounds obtained on (q)(n; k) are asymptotically tight in the sense that they imply lim k!1 (q)(n(k); k) (q) k = 0 for n(k) = b2 logq k 2 logq logq k f(k)c; and lim k!1 (q)(n(k); k) (q) k = 1 for n(k) = b2 logq k + f(k)c; where b:c denotes the integer part of the enclosed number. Finally, some approximations for (q)(n; k) are given that make the global behavior of (q)(n; k) more transparent.