Discrete Fourier Transform, Joint Linear Complexity and Generalized Joint Linear Complexity of Multisequences

Abstract

Let S ₁, S ₂, ..., S _t be t N-periodic sequences over ${\mathbb F}_{q}$. The joint linear complexity L(S ₁,S ₂,..., S _t ) is the least order of a linear recurrence relation that S ₁, S ₂, ...,S _t satisfy simultaneously. Since the ${\mathbb F}_{q}$-linear spaces ${\mathbb F}^{t}_{q}$ and ${\mathbb F}_{q^{t}}$ are isomorphic, a multisequence can also be identified with a single sequence ${\mathcal S}$ having its terms in the extension field ${\mathbb F}_{q^{t}}$. The linear complexity $L({\mathcal S})$ of ${\mathcal S}$, i.e. the length of the shortest recurrence relation with coefficients in ${\mathbb F}_{q^{t}}$ that ${\mathcal S}$ satisfies, may be significantly smaller than L(S ₁,S ₂,..., S _t ). We investigate relations between $L({\mathcal S})$ and L(S ₁,S ₂,..., S _t ), in particular we establish lower bounds on $L({\mathcal S})$ expressed in terms of L(S ₁,S ₂,..., S _t ).