Polynomial aspects of codes, matroids, and permutation by Cameron P.J. PDF

Polynomial aspects of codes, matroids, and permutation by Cameron P.J. PDF

By Cameron P.J.

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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 [7], 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. [2] 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).

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