By Cameron P.J.
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Extra resources for Polynomial aspects of codes, matroids, and permutation groups
4. Let (vi : i ∈ I) be vectors spanning a vector space V over F = GF(q). Verify the following description of the group of the code associated with the vector matroid defined by these vectors: • the domain is Ω = I × F; • the group is V ∗ ; • the action is given by f : (i, a) = (i, a + vi f ) for i ∈ I, a ∈ F and f ∈ V ∗ . CHAPTER 8 IBIS groups In this chapter, we consider a special class of permutation groups introduced by Cameron and Fon-Der-Flaass , which have a very close connection with matroids, in the sense that the bases for the permutation group form the bases of a matroid.
THis is because the function giving the ith coordinate of a vector is an element of the dual space, and these functions form a basis for the dual space. I leave as an exercise the problem of finding a matrix-free construction of the matroid from the code. 1 If the matroid M corresponds to the code C, then the dual matroid M ∗ corresponds to the dual code C⊥ . Proof If the matrix A happens to be in the form [Ik B], where Ik is a k × k identity matrix and B is k × n − k, then both the dual code and the dual matroid are represented by the matrix [−B In−k ].
3 For any finite permutation group G, we have FG (t) = PG (t + 1). Proof We know that PG (x) = Z(G; s1 ← x, si ← 1 for i > 1). Also, a set ∆ of cardinality n can be labelled in n! /|G(∆)| orbits under G. So we have n! t n FG (t) = ∑ ∑ n≥0 ∆∈P Ω/G,|∆|=n |G(∆)| n! 4. The Shift Theorem 53 the last equality coming from the Shift Theorem. So the result holds. However, the original proof by Boston et al.  is more direct. Let c1 (g) denote the number of fixed points of the element g. Then the number of ordered j-tuples of distinct elements it fixes is c1 (g)(c1 (g) − 1) · · · (c1 (g) − j + 1).