Linear complexity over F-q and over F-qm for linear recurring sequences

Since the F-q-linear spaces F-q(m) and F-qm are isomorphic, an m-fold multisequence S over the finite field F-q with a given characteristic polynomial f is an element of F-q[x], can be identified with a single sequence S over F-qm 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 oil 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. (C) 2008 Elsevier Inc. All rights reserved.