A New Family of $p$-Ary Sequences With Low Correlation Constructed From Decimated Sequences

Abstract

In this paper, for an odd prime <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex></formula> and an integer <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n \geq 3$</tex></formula> , a new family of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex> </formula> -ary sequences of period <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${{p^{n}-1} \over {2}}$</tex></formula> with low correlation is constructed. The new family is constructed by shifts and additions of two decimated sequences of a <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p$</tex></formula> -ary <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$m$</tex> </formula> -sequence, and its family size is <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$2(p^{n}-1)$</tex></formula> . The complete correlation distribution of this new family is derived. It is also shown that the family is optimal with respect to the parameter <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\theta_{{\rm rms}}$</tex></formula> , which denotes the root mean square of all nontrivial correlations. Compared with the known sequence families, our sequence family is new and has a larger family size, which is four times of its period.