Matroids and linear codes |

x not Îbar (A) and xÎbar (AÈ{y} ) then yÎbar (AÈ{x} ) " x,yÎX, AÍX A1)

Two matroids" AÍX$BÍX such that |B|<¥ and bar (B)=bar (A) A2)

Matroids which are representable over a fieldxÎbar (A)Ûf(x)Îbar (f(A)).

(GL_{k}(q)´GF(q)^{*}wr S_{n})´(GF(q)^{k})^{n}-> (GF(q)^{k})^{n}

where the monomial matrix((A,M),G) -> A·G·M,

For many parameter values *n*, *k* and *q* the numbers of isometry classes of
linear codes (with additional properties as "no repeated columns", "no
zero-columns" or "indecomposability") have been computed
[2][34] and in many cases there
are complete lists of representatives of these isometry classes available
[5].
See for instance

All these new results will be published in a forthcoming book [6] together with A. Kerber and other co-authors of Bayreuth and of Hamburg-Harburg. This book will be published by Springer (most of the details are already settled) and it will probably be the first text-book on coding theory describing how to get a complete overview over the isometry classes of linear codes. In other words in one part of this book we will describe all the details needed for enumerating these classes (combinatorics under finite group actions, Pólya theory, cycle index methods, especially the cycle indices of projective groups, ...) and for constructing complete lists of these codes (Sims chains, orderly generation combined with learning techniques, canonical forms, ...). Other parts of the book will give an introduction to error correcting codes and describe interesting connections between representation theory of groups and coding theory. In addition to this our software developed for the classification of linear codes will be made available by ftp.`/home/home_codes.html`

.

There are many interesting characterizations of regular matroids [1] and together with M. Wild [45][46] we designed and implemented a regularity test for binary matrix matroids in the computer algebra system SYMMETRICA [44]. This test takes as an input a binary matrix matroid and decides whether this matroid is regular or not. Since we have complete lists of the isomorphism classes of binary matrices we can produce complete lists of regular matrix matroids. This test seems to be the first regularity test for matroids which is completely implemented in a computer algebra system. When applying this regularity test on a matrix matroid we get some further information about it. For instance the sets of all hyperplanes co-lines and co-planes are computed or we can get an overview over all flats (closed sets) of a matroid.

Now it would be interesting to investigate the properties of these regular matroids. Determine whether there are connections between coding theoretic properties and matroidal properties. For instance indecomposable codes correspond to connected matroids, codes without repeated columns correspond to simple matroids, codes with no zero-columns correspond to loopless matroids. Since we have representatives of the isomorphism classes of regular matroids we really can put our hands on each of these objects and test them for certain prescribed properties.

For larger values of *n* and *k* we will use probabilistic methods like the *Dixon-Wilf-Algorithm* [11]
in order to construct binary or regular matrix
matroids distributed over all isomorphism classes uniformly at random, which
allows to produce huge sets of representatives and to check hypotheses
on them and afterwards try to prove the valid ones.

harald.fripertinger@kfunigraz.ac.at,

last changed: January 23, 2001

Matroids and linear codes |