Relative profiles and extended weight polynomials of almost affine codes

In this paper we study various aspects concerning almost affine codes, a class including, and strictly larger than, that of linear codes. We use the combinatorial tool demi-matroids to show how one can define relative length/dimension and dimension/length profiles of flags (chains) of almost affine codes. In addition we show two specific results. One such result is how one can express the relative length/dimension profiles (also called relative generalized Hamming weights) of a pair of codes in terms of intersection properties between the smallest of these codes and subcodes of the largest code. The other result tells how one can find the extended weight polynomials, expressing the number of codewords of each possible weight, for each code in an infinite hierarchy of extensions of a code over a given alphabet.