Recently six conjectures on linear complexities (LC) of some Kronecker sequences of two or three component sequences are proposed by Karkkainen. In, the LC of Kronecker sequences of two component sequences were studied by Uehara and Imamura, their results are true except in the case when eb 2 or when ea = eb = 1. In this paper the LC for Kronecker sequences of two component sequences are determined completely, and it is shown that all the six conjectures are true except in some special cases, which are listed and corrected.

We introduce a graphical calculus for multi-qutrit systems (the qutrit ZX-calculus) based on the framework of dagger symmetric monoidal categories. This graphical calculus consists of generators for building diagrams and rules for transforming diagrams, which is obviously different from the qubit ZX-calculus. As an application of the qutrit ZX-calculus, we give a graphical description of a (2, 3) threshold quantum secret sharing scheme. In this way, we prove the correctness of the secret sharing scheme in a intuitively clear manner instead of complicated linear algebraic operations.