Extended combinatorial algebraic approach for the second-generation time-delay interferometry

This work elaborates on an algebraic approach to second-generation time-delay interferometry (TDI). The proposed method is closely related to the algorithm first developed by Dhurandhar et al. and its recent generalizations. While the relevant equation is derived from a geometric TDI perspective, the resulting TDI solutions are primarily generated by a basis consisting of four-tuples. Unlike the original study, the present scheme is not subject to any constraint equation and spans the underlying solution space much further. Moreover, the algorithm does not rely on specific subscript permutations regarding the two elements of the commutator that furnishes the TDI solution. Employing the proposed method, we explicitly show that all the existing second-generation TDI combinations, most established via the geometric TDI approach, can be derived. It is argued that the current approach provides an alternative perspective on the algebraic structure of the second-generation TDI solutions.