Some Upper Bounds and Exact Values on Linear Complexities Over F<sub> <i>M</i> </sub> of Sidelnikov Sequences for <i>M</i> = 2 and 3

Sidelnikov sequences, a kind of cyclotomic sequences with many desired properties such as low correlation and variable alphabet sizes, can be employed to construct a polyphase sequence family that has many applications in high-speed data communications. Recently, cyclotomic numbers have been used to investigate the linear complexity of Sidelnikov sequences, mainly about binary ones, although the limitation on the orders of the available cyclotomic numbers makes it difficult. This paper continues to study the linear complexity over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{M}$ </tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> -ary Sidelnikov sequence of period <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q-1$ </tex-math></inline-formula> using Hasse derivative, which implies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=p^{m}$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq 1$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M|(q-1)$ </tex-math></inline-formula> . The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> th Hasse derivative formulas are presented in terms of cyclotomic numbers, and some upper bounds on the linear complexity for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$M=2$ </tex-math></inline-formula> and 3 are obtained only with some additional restrictions on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> . Furthermore, concrete illustrations for several families of these sequences, such as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\equiv 1\pmod {2}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q\equiv 1\pmod {3}$ </tex-math></inline-formula> , show these upper bounds are tight and reachable; especially for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=2\times 3^{\lambda }+1 (1\leq \lambda \leq 20)$ </tex-math></inline-formula> , the exact linear complexities over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{3}$ </tex-math></inline-formula> of the ternary Sidelnikov sequences are determined; and it turns out that all the linear complexities of the sequences considered are very close to their periods.