Abstract: The ordered Hamming space (Niederreiter-Rosenbloom-Tsfasman space) was introduced independently and for different applications by Niederreiter, and by Rosenbloom and Tsfasman. We will describe the ordered Hamming space and then present new bounds - an Elias-type bound and Linear Programming bounds - on the size of codes and orthogonal arrays in that space. Joint work with Alexander Barg.