<p>The construction of state-of-the-art codes based on several types of combinatorial designs has attracted some attention. The application of directed designs and their corresponding trade designs motivated us to investigate the necessary and sufficient conditions to have super-simple directed group divisible designs with block size 4 in which each ordered pair of elements, that are not in a group, appears in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\lambda =1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> block (or simply (4,&#xa0;1)-DGDD). We prove that the necessary condition for these designs is also sufficient. The defining set related to most of our proposed designs contains at least half of the blocks. Moreover, we provide a number of super-simple (4,&#xa0;2)-DGDDs whose defining sets have at least half of the blocks. These designs are corrected versions of their counterparts in the literature. All proposed designs for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\lambda =1,2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> can be used to develop LDPC (low-density parity-check) codes of current practical interest.</p>