The multivariable case of Finite Settling Time Stabilisation ( FSTS) of linear discrete-time systems is considered in this paper. An algebraic approach is adopted which leads to the solution of a polynomial matrix Diophantine equation. This gives rise to the parametrisation of all FSTS controllers in a Kucera-Youla-Bonjiorno sense and the FSTS problem is further reduced to a linear algebra problem over the real numbers. Subsequently, the family of all causal FSTS controllers is parametrised, and necessary and sufficient conditions for strong FSTS ( stable controllers) are derived. The minimal McMillan degree solution and minimal complexity controllers are examined and new bounds are given. The analysis provides the means for the parametrisation of families of FSTS controllers with certain complexity. Finally the problems of tracking, disturbance rejection and partially assigned dynamics in FST sense are considered and conditions for their solvability are given.