Projective de Bruijn Sequences

Abstract

Let ${{\mathbb F}_q}$ denote the q-element field. A q-ary de Bruijn sequence of degree m is a cyclic sequence of elements of ${{\mathbb F}_q}$, such that every element in ${{\mathbb F}_q}^m$ appears exactly once as a consecutive m-tuple in the cyclic sequence. We consider its projective analogue; namely, a cyclic sequence such that every point in the projective space $({{\mathbb F}_q}^{m+1} -\{0\})/({{\mathbb F}_q}^\times)$ appears exactly once as a consecutive (m + 1)-tuple. We have an explicit formula $(q!)^{\frac{q^m-1}{q-1}}q^{-m}$ for the number of distinct such sequences.

This work is supported in part by JSPS Grant-In-Aid #16204002, #18654021, #18740044, #19204002 and JSPS Core-to-Core Program No.18005.