A general secondary construction of Boolean functions including the indirect sum and its generalizations

We study a secondary construction of Boolean functions, which generalizes the direct sum and the indirect sum. We detail how these two classic secondary constructions are particular cases of this more general one, as well as two known generalizations of the indirect sum. This unifies the known secondary constructions of Boolean functions. We study very precisely the Walsh transform of the constructed functions. This leads us to an interesting observation on the Walsh transforms $W_g,W_{g^{\mathrm{\prime }}},W_{g^{\mathrm{\prime }\mathrm{\prime }}}$, and $W_{g\oplus g^{\mathrm{\prime }}\oplus g^{\mathrm{\prime }\mathrm{\prime }}}$ when $g,g^{\mathrm{\prime }},g^{\mathrm{\prime }\mathrm{\prime }}$ are Boolean functions such that $(g\oplus g^{\mathrm{\prime }})(g\oplus g^{\mathrm{\prime }\mathrm{\prime }})$ equals the zero function.