Relative generalized hamming weights and extended weight polynomials of almost affine codes

This is a post-peer-review, pre-copyedit version of an article published in Lecture Notes in Computer Science, International Castle Meeting on Coding Theory and Applications ICMCTA 2017: Coding Theory and Applications, 207-216. The final authenticated version is available online at: http://dx.doi.org/10.1007/978-3-319-66278-7_17 .This paper is devoted to giving a generalization from linear codes to the larger class of almost affine codes of two different results. One such result is how one can express the 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. Our tools will be demi-matroids and matroids

Almost affine codes and matroids

In this thesis we study various types of block codes, like linear, mutlti-linear, almost affine codes. We also look at how these codes can be described by associated matroids. In addition we look at flags (chains) of codes and see how their behavior can be described using demi-matroids. We also introduce weight polynomials for almost affine codes