✝ ☎ ☎ ✝ ✝ ☎ ☎ ☎ ☎ ☎ ☎ ✆ ✝ ✝ ✝ ✝ ✝ ✝ ☎ ☎ ✝ ✖ ✂ ✁ � ☎ ☎ ✆ ✜ ✄ ✜ ✜ Codes Polynomials associated with ☛ code over GF ✠ n ✡ k ☞ q ✌ is a k -dimensional subspace An ✌ n . Its elements are called codewords . of GF ☞ q permutation groups, The weight wt matroids and codes ☞ v ✌ of v is the number of non-zero coordinates of v . The weight enumerator of C is the polynomial Peter J Cameron ✓ Y wt ∑ X n ✑ wt ✒ v ✒ v W C ☞ X ✡ Y ✓✕✔ School of Mathematical Sciences ✌✎✍ v ✏ C Queen Mary, University of London London E1 4NS, U.K. The weight enumerator of a code carries a lot of p.j.cameron@qmul.ac.uk information about it; but different codes can have the same weight enumerator. 1 3 Matroids A map A matroid on a set E is a family of subsets of E (called independent sets with the properties ✗ a subset of an independent set is independent; ✗ if A and B are independent with ✘ A ✘✚✙✛✘ B ✘ , then there exists x B ✢ A such that A ✣✥✤ x ✦ is independent. ? ? ? ? ? The rank ρ ☞ A ✌ of a subset A of E is the common size ? ? of maximal independent subsets of A . ✝✟✞ Examples of matroids: ✗ E is a family of vectors in a vector space, Permutation groups Matroids Tutte polynomial Cycle index independence is linear independence; ✗ E is a family of elements in a field K , independence ✝✟✞ is algebraic independence over a subfield F ; ✗ E is the set of edges of a graph, a set is independent if it is acyclic; Codes ✗ E is the index set of a family ☞ A i : i E Weight enumerator ✌ of subsets ☞ A i : i of X , a set I is independent if I ✌ has a system of distinct representatives. 2 4
✯ ✯ ✍ ✰ ✯ ✮ ✴ ✯ ✰ ✯ ✌ ✴ ✯ ✔ ✡ ✓ ✓ ✡ ✡ ✦ ✌ ✜ ✯ ✔ ✲ ✯ ✩ ✡ ✰ ✰ ✓ ✔ ✡ ✓ ✯ ✫ ✯ ✩ ✩ ✌ Permutation groups Tutte polynomial Let G be a permutation group on E , that is, a subgroup of the symmetric group on E , where n . The cycle index of G is the polynomial Z The Tutte polynomial of a matroid M is given by ✘ E ☞ G ✘✬✍ in indeterminates s 1 ✡ s n given by ✔✕✔✕✔ ∑ ✌ ρ ✒ E ✓✪✑ ρ ✒ A ✌✬✫ A ✑ ρ ✒ A ☞ M ; x 1 1 T ✡ y ☞ x ☞ y ✌✧✍ 1 s c 1 ✒ g ✳✕✳✕✳ s c n ✒ g ✘ ∑ A ★ E Z ☞ G n 1 ✌✧✍ ✘ G where ρ is the rank function of M . g ✏ G In particular, The Tutte polynomial carries a lot of information 1 for i 1 P G ☞ x Z ☞ G ✌✬☞ s 1 x ✡ s i ✌✧✍ ☞ M ;2 ✡ 1 about the matroid; e.g. T ✌ is the number of is the p.g.f. for the number of fixed points of a random independent sets, and T ☞ M ;1 ✡ 1 ✌ is the number of element of G . bases (maximal independent sets). But there exist different matroids with the same Tutte polynomial. The cycle index is very important in enumeration theory. Two simple examples: ✗ Z 1 1 for i 1 ☞ G ✌✬☞ s 1 x ✡ s i The Tutte polynomial of a matroid generalises the ✌ is the exponential generating function for the number of G -orbits on Jones polynomial of a knot, percolation polynomials, k -tuples of distinct points (note that this function is etc.; and also the weight enumerator of a code, as P G ☞ x 1 ✌ ); we will see. ✗ Z x i 1 ☞ G ✌✬☞ s i ✌ is the ordinary generating function for the number of orbits of G on k -subsets of E . 5 7 Permutation groups and codes Matroids and codes ☛ code over GF Let C be an ✠ n ✡ k ☞ q ✌ . The additive group G of C acts as a permutation group on the set ✠ n ✡ k ☛ code C we may associate in a With a linear GF ✌✶✵✷✤ 1 E ☞ q ✡ n ✔✕✔✕✔ ✦ by the rule that the codeword ✤ 1 canonical way a matroid M C on the set ✡ n ✔✕✔✕✔ v ✍✸☞ v 1 ✡ v n ✔✕✔✕✔ ✌ acts as the permutation whose independent sets are the sets I for which the ☞ c i : i I ✌ of a generator matrix for C are columns ☞ x ✡ i ✌✧✹✺✻☞ x v i ✡ i linearly independent. Now each permutation has cycles of length 1 and p Curtis Greene showed that the weight enumerator of only, where p is the characteristic of GF ☞ q ✌ ; and we the code is a specialisation of the Tutte polynomial of have the matroid: 1 X 1 ✼ q Y p ✼ q ☞ G ; s 1 1 ✘ W C ☞ X ✡ Y Z ✡ s p X ✰✱☞ q ✌ Y X Y n ✑ k ✌ k T ✌✎✍ ✌✽✡ M C ; x ✘ C W C ☞ X ✡ Y ☞ X ✩ Y ✡ y ✌✭✍ X ✩ Y Y For a zero coordinate in v gives rise to q fixed points, and a non-zero coordinate to q ✾ p cycles of length p . I use the notation F ☞ x t ✌ to denote the result of substituting the term t for x in the polynomial F . So the cycle index of G carries the same information as the weight enumerator of C . 6 8
☛ ✰ ✡ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ ✰ Base-transitive groups IBIS groups A permutation group is base-transitive if it permutes Let G be a permutation group on Ω . A base for G is a its irredundant bases transitively. A base-transitive sequence of points of Ω whose stabiliser is the group is clearly an IBIS group. identity. It is irredundant if no point in the sequence is All base-transitive groups of rank at least 2 have fixed by the stabiliser of its predecessors. been determined by Maund, using CFSG; those of large rank (at least 7 ) by Zil’ber, by a geometric Cameron and Fon-Der-Flaass showed that the argument not using CFSG. following three conditions on a permutation group are equivalent: The matroid associated with a base-transitive group is a perfect matroid design ; this is a matroid of rank r ✗ all irredundant bases have the same number of for which the cardinality n i of an i -flat (a maximal set points; of rank i ) depends only on i . ✗ re-ordering any irredundant base gives an Mphako showed that the Tutte polynomial of a PMD irredundant base; is determined by the cardinalities n 1 ✡ n r of its flats. ✗ the irredundant bases are the bases of a matroid. ✔✕✔✕✔ If the matroid arises from a base-transitive group, these are the numbers of fixed points of group A permutation group satisfying these conditions is elements. Thus, for a base-transitive group, the cycle called an IBIS group (short for Irredundant Bases of index determines the Tutte polynomial of the matroid, Invariant Size). but not conversely. 9 11 An example Examples of IBIS groups The cycle index does not in general tell us whether a permutation group is base-transitive. The groups ✗ Any Frobenius group is an IBIS group of rank 2 , ✍✿✤ 1 ✡❀☞ 1 ✡ 2 ✌✬☞ 3 ✡ 4 ✌✽✡❁☞ 1 ✡ 3 ✌✬☞ 2 ✡ 4 ✌✬✡❀☞ 1 ✡ 4 ✌✬☞ 2 ✡ 3 G 1 ✌❁✦❂✡ associated with the uniform matroid. ✍✿✤ 1 ✡❀☞ 1 ✡ 2 ✌✬☞ 3 ✡ 4 ✌✽✡❁☞ 1 ✡ 2 ✌✬☞ 5 ✡ 6 ✌✬✡❀☞ 3 ✡ 4 ✌✬☞ 5 ✡ 6 G 2 ✗ The general linear and symplectic groups, acting on ✌❁✦ of degree 6 have the same cycle index, namely their natural vector spaces, are IBIS groups, 1 ☞ s 6 3 s 2 1 s 2 Z ☞ G ✌ . The first is base-transitive with 4 1 2 associated with the vector matroid (defined by all ✌✧✍ rank 1 ; the second is an IBIS group of rank 2 (arising vectors in the space). from the binary even-weight code of length 3 ). ✗ The Mathieu group M 24 in its natural action is an IBIS group of rank 7 . If a group with this cycle index is base-transitive then ✗ The permutation group constructed from an ✠ n ✡ k Mphako’s result gives the Tutte polynomial as linear code over GF y 2 ☞ y 3 y 2 ☞ q ✌ is an IBIS group of degree nq y x ✌ . and rank k . The associated matroid is obtained from If a group with this cycle index comes from a code, the matroid of the code simply by replacing each element by a set of q parallel elements. It is we can calculate the Tutte polynomial to be y 4 2 y 3 3 y 2 x 2 3 xy y x . straightforward to obtain the Tutte polynomial of the group matroid from that of the code matroid and vice In the second case, the matroid admits two different versa . base-transitive groups with different cycle indices (both isomorphic to S 4 as abstract groups). 10 12
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