Linear complexity over [formula omitted] and over [formula omitted] for linear recurring sequences

Abstract

Since the F q -linear spaces F q m 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.