We report the growth by molecular beam epitaxy (MBE) of InAlSb/InSb superlattice structures to investigate their potential for reducing the Auger recombination and intervalence absorption effects, which currently limit the maximum operating temperature. The devices were all grown onto InSb(001) substrates and are not lattice matched. They are a psuedo double heterostructure, comprising an active region 3micrometers thick, consisting of 10 repeats of In0.904Al0.096Sb/InSb (10nm/6.5nm), surrounded by undoped layers of In0.944Al0.056Sb. Electrical confinement in the active region is by means of a 20nm thick wide gap In0.794Al0.206Sb barrier layer onto which a p type In0.944Al0.056Sb tunnel contact and highly doped n type In0.944Al0.056Sb region is grown, which together with a substrate highly doped n type In0.944Al0.056Sb region gives optical confinement, due to a Moss-Burstein shift of the refractive index. We have demonstrated laser operation up to 160K for devices ~1000micrometers long by 15micrometers wide. FTIR spectroscopy measurements revealed a lasing wavelength of 3.65micrometers at 80K. Under pulse bias conditions, the threshold current density was 320Acm-2 at 80K. The peak output power was in excess of 800mW. Detailed modeling of the structures shows that greater strain is required in the system in order to quench Auger losses at higher temperatures.