Mixed-level designs are widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. This paper investigates the sufficient and necessary conditions for a <InlineEquation ID="IEq1"> <EquationSource...</equationsource></inlineequation>

Zhang et al. (Stat Sinica 18:1689–1705, <CitationRef CitationID="CR15">2008</CitationRef>) introduced an aliased effect-number pattern for two-level regular designs and proposed a general minimum lower-order confounding (GMC) criterion for choosing optimal designs. All the GMC <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$2^{n-m}$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>-</mo> <mi>m</mi> </mrow> </msup> </math> </EquationSource> </InlineEquation> designs with <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$N/4+1\le n\le...</equationsource></inlineequation></equationsource></equationsource></inlineequation></citationref>