New $M$-Ary Sequence Families With Low Correlation and Large Size

Abstract

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we construct four <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis>-ary sequence families from a <emphasis emphasistype="boldital">power residue sequence</emphasis> of odd prime period <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$p$</tex></formula></emphasis> and its constant multiple sequences using the shift-and-add method, when <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis> is a divisor of <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$p-1$</tex></formula></emphasis>. We show that the maximum correlation values of the proposed sequence families are upper-bounded by <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$2\sqrt {p} +5$</tex></formula></emphasis> or <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$3\sqrt {p} +4$</tex></formula></emphasis>. In addition, we prove that the linear complexity of each sequence in the proposed families is either <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$p\!-\!1$</tex></formula></emphasis> or <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$p\!-\!{p-1 \over M}\!-\!1$</tex> </formula></emphasis>. We also construct an <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex></formula></emphasis>-ary sequence family from <emphasis emphasistype="boldital">Sidel'nikov sequences</emphasis> of period <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$p^m-1$</tex></formula></emphasis> by applying the same method, when <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$M$</tex> </formula></emphasis> is a divisor of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$p^m-1$</tex></formula></emphasis>. The proposed sequence family <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$\mathtilde {\cal F}_{\mmb s}$</tex></formula></emphasis> has larger size than the known <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$M$</tex></formula></emphasis>-ary Sidel'nikov sequence families, whereas they all have the same upper bound on the maximum correlation. </para>