ARTICLE IN PRESS YFFTA:697 Please cite this article in press as: W. Meidl, F. Özbudak, Linear complexity over F q and over F q m for linear recurring sequences, Finite Fields Appl. (2008), doi:10.1016/j.ffa.2008.09.004 JID:YFFTA AID:697 /FLA [m1G; v 1.66; Prn:16/10/2008; 9:44] P.1 (1-15) Finite Fields and Their Applications ••• (••••) •••–••• Contents lists available at ScienceDirect Finite Fields and Their Applications www.elsevier.com/locate/ffa Linear complexity over F q and over F q m for linear recurring sequences Wilfried Meidl a , Ferruh Özbudak b,∗ a Faculty of Engineering and Natural Sciences, Sabancı University, Tuzla, 34956, ˙ Istanbul, Turkey b Department of Mathematics, Middle East Technical University, ˙ Inönü Bulvarı, 06531, Ankara, Turkey article info abstract Article history: Received 7 August 2008 Revised 25 September 2008 Communicated by Gary L. Mullen Keywords: Joint linear complexity Generalized joint linear complexity Multisequences Linear recurring sequences Since the F q -linear spaces F m q and F q m are isomorphic, an m-fold multisequence S over the finite field F q with a given characteristic polynomial f ∈ F q [x], can be identified with a single sequence S over F q m with characteristic polynomial f . The linear complexity of S, which will be called the generalized joint linear complexity of S , can be significantly smaller than the conventional joint linear complexity of S . We determine the expected value and the variance of the generalized joint linear complexity of a random m- fold multisequence S with given minimal polynomial. The result on the expected value generalizes a previous result on periodic m- fold multisequences. Moreover we determine the expected drop of linear complexity of a random m-fold multisequence with given characteristic polynomial f , when one switches from conventional joint linear complexity to generalized joint linear complexity.  2008 Elsevier Inc. All rights reserved. 1. Introduction A sequence S = s 0 , s 1 ,... with terms in a finite field F q with q elements (or over the finite field F q ) is called a linear recurring sequence over F q with characteristic polynomial f (x) = l  i=0 c i x i ∈ F q [x] * Corresponding author. E-mail addresses: wmeidl@sabanciuniv.edu (W. Meidl), ozbudak@metu.edu.tr (F. Özbudak). 1071-5797/$ – see front matter  2008 Elsevier Inc. All rights reserved. doi:10.1016/j.ffa.2008.09.004