Is cubic complexity still the state of the art for LP?
It seems that K.M.Anstreicher has improved the result to $O((n^3/\ln n)L)$ in Anstreicher, Kurt M. "Linear programming in O ([n3/ln n] L) operations." SIAM Journal on Optimization 9, no. 4 (1999): 803-812.. I have not read this paper, but I hope that this answer will help you in some extent.