The Index of an M -Sequence
Michael S. Willett · SIAM Journal on Applied Mathematics · 1973
Let $( {u_i } )_0^\infty $ be an m-sequence of period $q^n - 1$ over $GF( q )$, and let $I_0 = \{ i| 0\leqq i< ( q^{n - 1} ) / ( q - 1),u_i = 0 \}$. Our main result is that the index of u, denoted by index $( u )$, satisfies \[ q \cdot \operatorname{index} ( u ) \equiv ( {q - 1} )\{ {a - q\sum\limits_{i \in I_0 } i } \}\bmod ( {q^n - 1} ), \] where $a = 0$ or $( q^n - 1 )/2( q - 1 )$, if $( q^n - 1 )/( q - 1 )$ is odd or even respectively.