Slide 1 Slide 3 Codes Finite geometry and permutation groups: An [ n , k ] code over GF ( q ) is a k -dimensional sub- space of GF ( q ) n . Its elements are called codewords . some polynomial links The weight wt ( v ) of v is the number of non-zero coordinates of v . The weight enumerator of C is the Peter J Cameron polynomial W C ( X , Y ) = ∑ X n − wt ( v ) Y wt ( v ) . v ∈ C p.j.cameron@qmul.ac.uk The weight enumerator of a code carries a lot of in- formation about it; but different codes can have the Trends in Geometry same weight enumerator. Roma, June 2004 Slide 4 Matroids Slide 2 A matroid on a set E is a family I of subsets of E A map (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 indepen- dent. ? ? 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, indepen- � ✒ ❅ ■ � ❅ dence is linear independence; � ❅ • E is a family of vectors in a vector space, indepen- dence is affine independence; Permutation groups Matroids • E is a family of elements in a field K , indepen- Tutte polynomial Cycle index dence is algebraic independence over a subfield F ; ❅ ■ ✒ � • E is the set of edges of a graph, a set is indepen- ❅ � dent if it is acyclic; ❅ � • E is the index set of a family ( A i : i ∈ E ) of sub- sets of X , a set I is independent if ( A i : i ∈ I ) has a Codes Weight enumerator system of distinct representatives. 1
Slide 5 Slide 7 Matroids and finite geometry Matroids and codes Specialising the first example above, we see that any With a linear [ n , k ] code C we may associate in a canonical way a matroid M C on the set { 1 ,..., n } set of points in a finite projective space gives rise to a matroid, which captures a lot of the geometric whose independent sets are the sets I for which the columns ( c i : i ∈ I ) of a generator matrix for C are properties of the set. In particular, Segre’s fundamental problem about linearly independent. the size and classification of arcs in PG ( k , q ) is Curtis Greene showed that the weight enumerator of equivalent to the problem of classifying represen- the code is a specialisation of the Tutte polynomial tations of the uniform matroid U k + 1 , n (whose bases of the matroid: are all ( k + 1 ) -subsets of an n -set) over GF ( q ) . The � M C ; x ← X +( q − 1 ) Y � , y ← X W C ( X , Y ) = Y n − k ( X − Y ) k T coding theory version of this problem is the classi- . X − Y Y fication of the maximum distance separable codes over GF ( q ) . I use the notation F ( x ← t ) to denote the result of substituting the term t for x in the polynomial F . Slide 8 Permutation groups Let G be a permutation group on E , that is, a sub- group of the symmetric group on E , where | E | = n . The cycle index of G is the polynomial Z ( G ) in in- determinates s 1 ,..., s n given by Slide 6 Z ( G ) = 1 | G | ∑ s c 1 ( g ) ··· s c n ( g ) . n 1 Tutte polynomial g ∈ G In particular, The Tutte polynomial of a matroid M is given by T ( M ; x , y ) = ∑ ( x − 1 ) ρ ( E ) − ρ ( A ) ( y − 1 ) | A |− ρ ( A ) , P G ( x ) = Z ( G )( s 1 ← x , s i ← 1 for i > 1 ) A ⊆ E is the p.g.f. for the number of fixed points of a ran- where ρ is the rank function of M . dom element of G . The Tutte polynomial carries a lot of information The cycle index is very important in enumeration about the matroid; e.g. T ( M ;2 , 1 ) is the number of theory. Two simple examples: independent sets, and T ( M ;1 , 1 ) is the number of • Z ( G )( s 1 ← x + 1 , s i ← 1 for i > 1 ) is the exponen- bases (maximal independent sets). But there exist tial generating function for the number of G -orbits different matroids with the same Tutte polynomial. on k -tuples of distinct points (note that this function The Tutte polynomial of a matroid generalises the is P G ( x + 1 ) ); • Z ( G )( s i ← x i + 1 ) is the ordinary generating func- Jones polynomial of a knot, percolation polynomi- als, etc.; and also the weight enumerator of a code, tion for the number of orbits of G on k -subsets of as we will see. E . 2
Slide 9 Slide 11 The Shift Theorem Base-transitive groups We require the Shift Theorem : A base for a permutation group is a sequence of points whose pointwise stabiliser is the identity. A ∑ Z ( G ; s i ← s i + 1 ) = Z ( G ( A )) , base is irredundant if no point is fixed bu the point- A ∈ P E / G wise stabiliser of its predecessors. A permutation group is base-transitive if it permutes where E = { 1 ,..., n } , P E / G denotes a set of orbit its irredundant bases transitively. In this case, the representatives for G acting on the power set P E of irredundant bases are the bases of a matroid, indeed E , and G ( A ) is the permutation group induced on A a perfect matroid design ; this is a matroid of rank r by its setwise stabiliser G A in G . for which the cardinality n i of an i -flat (a maximal For example, if we sum the cycle indices of the sym- set of rank i ) depends only on i . In this case the Tutte metric groups of degree k for k = 0 , 1 ,..., n , then we polynomial is determined by the numbers n 0 ,..., n r . obtain Z ( S n ) with the substitution s i ← s i + 1. All base-transitive groups of rank at least 2 have been determined by Maund, using CFSG; those of large rank (at least 7) by Zil’ber, by a geometric ar- gument not using CFSG. For base-transitive groups, the cycle index deter- mines the cardinalities of the flats, and hence the Tutte polynomial, but not conversely. Slide 10 Permutation groups and codes Let C be an [ n , k ] code over GF ( q ) . The additive group G of C acts as a permutation group on the set Slide 12 E = GF ( q ) × { 1 ,..., n } by the rule that the code- word v = ( v 1 ,..., v n ) acts as the permutation The main problem ( x , i ) �→ ( x + v i , i ) . As we have seen, there are cases when the Tutte polynomial determines the cycle index (groups from Now each permutation has cycles of length 1 and p codes), and cases where the cycle index determines only, where p is the characteristic of GF ( q ) ; and we the Tutte polynomial (base-transitive groups). have Is there a more general polynomial which deter- mines both? 1 | C | W C ( X , Y ) = Z ( G ; s 1 ← X 1 / q , s p ← Y p / q ) , The situation we will take is a matroid M and a group G of automorphisms of M . For a zero coordinate in v gives rise to q fixed points, We would like this polynomial to specialise to allow and a non-zero coordinate to q / p cycles of length p . us to count orbits of G on configurations enumer- So the cycle index of G carries the same information ated by the Tutte polynomial of M (such as bases or as the weight enumerator of C , and is determined by independent sets, or coefficients of the weight enu- the Tutte polynomial. merator of a code). 3
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