Permutations and codes: set of words. Polynomials, bases, and - PDF document

✒ ✁ ✌ ✎ ✡ ✁ ✁ ✡ Hamming distance Hamming distance d ✟ is the number of coordinate ✞ x ✄ y positions where two words differ. It is a metric on the Permutations and codes: set of words. Polynomials, bases, and covering radius In the binary case, ✟✠✁ ✟☛✄ d ✞ x ✄ y wt ✞ x y Peter J. Cameron so for a linear code, minimum distance equals Queen Mary, University of London smallest number of non-zero coordinates of a p.j.cameron@qmw.ac.uk non-zero element ( minimum weight ). International Conference on Graph Theory In the permutation group case, Bled, 22–27 June 2003 ✟✠✁ ✟☛✄ ☞ 1 y d ✞ x ✄ y n fix ✞ x so, for a permutation group, minimum distance equals smallest number of points moved by a non-identity element ( minimal degree ). 1 3 Binary codes and sets of permutations An apology We will be considering sets of n -tuples over an This is not really graph theory: the distance between alphabet A , in two important cases: permutations is not a graph distance, because there ✂ 0 do not exist two permutations at distance 1 . � A ☎ (binary code); ✄ 1 ✂ 1 However, it is closely related to the distance d ✌ in the � A ✄✝✆✝✆✝✆✝✄ n ☎ , all entries of each word distinct Cayley graph of the symmetric group with respect to (set of permutations). the set of transpositions: we have ✟✝✍ 2 ✟✏✎ ✟✑✡ We often impose closure conditions on these sets, as d ✞ g ✄ h d ✞ g ✄ h d ✞ g ✄ h 1 follows: for g h . � A binary code is linear if it is closed under Also, we will be considering the size of the smallest coordinatewise addition mod 2 . dominating set in the graph G n ✓ k with vertex set S n , two permutations joined if they agree in at least k � A set of permutations is a group if it is closed places. under composition. 2 4

Bases for permutation groups Some analogies Let G be a permutation groups. A base is a sequence of points whose pointwise stabiliser is the Linear code Permutation group identity. It is irredundant if no point is fixed by the length degree stabiliser of its predecessors, and is minimal if no minimum weight minimal degree point is fixed by the stabiliser of all the others. weight enumerator permutation character Tutte polynomial cycle index Note that changing the order preserves the basis base dimension base size properties of being a base and minimality, but not covering radius ?? necessarily irredundance. However, computationally it is easy to produce an irredundant base but much harder to find a minimal base. 5 7 Groups as codes IBIS groups The idea of coding with permutations goes back to Blake, Cohen and Deza in the 1970s. The following are equivalent for the permutation group G : Among their suggestions was that the Mathieu group M 12 would be a good code (comparable to a � all irredundant bases have the same number of Reed–Solomon code). It has minimal degree 8 , so is elements; 3 -error-correcting. � the irredundant bases are preserved by Recently, R. F . Bailey showed that it corrects about 96% of all four-error patterns. reordering; � the irredundant bases are the bases of a matroid. Also, it is easy to decode, using efficient algorithms for permutation groups. Bailey’s decoding algorithm A group with these properties is called an IBIS group uses a covering design to give a collection of 5 -sets ( I rredundant B ases of I nvariant S ize). such that at least one is disjoint from each error pattern. Then find the unique element of M 12 agreeing with the received word on that 5 -set. 6 8

✁ ✙ ✁ ✁ ✁ ✙ ✡ ✁ ✙ Examples of IBIS groups The Tutte cycle index Any linear code C of length n gives an IBIS group The Tutte cycle index of a permutation group G on Ω with essentially the same matroid, as follows. The set ✂ 1 ✄✝✆✝✆✝✆✝✄ n ☎✕✔ GF is of points permuted is ✟ ; the group is ✞ 2 the additive group of C ; the action is ✚ ∑ ✤✦✥ Z 1 ✟✠✁ ✣ ∆ ★✩✟☛✆ ✜ v b ✢ G ✜ G ∆ ✧ ∆ ✟✠✖✗✘✞ i ✟☛✆ ✚ G ZT ✞ G u ✞ G c : ✞ i ✄ x ✄ x c i ∆ ✛ Ω The matroid is just the usual matroid of the code with each element ‘doubled’. Here G ∆ is the setwise stabiliser of ∆ , G ✥ its ✢ ∆ ✧ ∆ ★✫✪ pointwise stabiliser, G ✥ the group G ∆ ✍ G ✢ ∆ There are many other examples: symmetric and induced on ∆ by G ∆ , and b is the minimum base size. alternating groups; linear and affine groups; linear fractional groups; and many sporadic ones. To get the cycle index: differentiate with respect to u , put u 1 , and replace s i by s i 1 . v A group which permutes its irredundant bases transitively is an IBIS group. Such groups were To get the Tutte polynomial (if G is IBIS): put u 1 determined by Maund (using the Classification of t i . (The result is actually T ✟ .) and s i ✞ M ; v ✍ t 1 ✄ t 1 Finite Simple Groups). 9 11 The geometry of bases We have seen that, in an IBIS group, the irredundant Polynomials (or minimal) bases satisfy the matroid basis axioms. A permutation group has a cycle index polynomial. If What kind of configuration do they form in more it is an IBIS group, it is associated with a matroid, general cases (when the two kinds may not which has a Tutte polynomial . coincide)? In particular, we may ask this question in two special cases: Sometimes (e.g. for the groups obtained from linear � What if all minimal bases have the same codes) the cycle index is a specialisation of the Tutte polynomial; sometimes (e.g. for base-transitive cardinality? groups) it is the other way round. � What if the irredundant bases have cardinalities differing by one? It is possible to define a more general polynomial, the Tutte cycle index , which specialises to the cycle index The greedy algorithm produces an irredundant base and (in the case of an IBIS group) also to the Tutte by choosing each base point to lie in an orbit of polynomial. Its properties haven’t been investigated largest size of the stabiliser of its predecessors. We systematically. can also ask whether the greedy bases (produced in this way) have nicer properties than arbitrary irredundant bases. 10 12

✎ ✁ ✆ ✙ ✙ ✆ ✎ ✁ ✚ ✎ ✁ ☎ Base sizes An example: S 5 on pairs Imre Leader asked: An example satisfying both conditions on the preceding slide is the symmetric group S 5 acting on the ten edges of the complete graph K 5 . Do the base sizes of a permutation group form an interval? A minimal base consists of the three edges of a forest with one isolated vertex and one 4 -vertex tree. The answer is ‘no’ for minimal bases. The group C 3 2 , The minimal bases are not the bases of a matroid in ✂ 12 ✂ 12 with three orbits of size 2 and one regular orbit of this case. (For B ☎ is a base; I ✄ 23 ✄ 34 ✄ 45 size 8 , has minimal bases of size 1 (a point in the is contained in a base, but it is not possible to add an regular orbit) and 3 (one point in each orbit of size 2 ) element of B to it to form a base.) Note that the only. permutation group bases are some of the bases of the cycle matroid of K 5 truncated to rank 3 . However, it is true for irredundant bases: if a group has irredundant bases of sizes m 1 and m 2 , then it has The 4 -tuple ✞ 12 ✄ 45 ✄ 23 ✄ 34 ✟ is an irredundant base irredundant bases of all intermediate sizes. which is not minimal. What happens for greedy bases? (Note that the The greedy algorithm always produces a minimal greedy algorithm is not deterministic since at some base in this example. (This is not always the case!) point there may be several largest orbits.) 13 15 Base sizes The largest irredundant base has size at most log 2 n Covering radius times that of the smallest. Indeed, if G has an irredundant base of size b , then ✚ G Let S be a subset of a finite metric space M . The 2 b n b packing radius of S is the maximum r such that the balls of radius r with centres at points of S are The largest base chosen by the greedy algorithm has pairwise disjoint; the covering radius is the minimum size at most ✟ times that of the smallest ✞ loglog n c R such that the balls of radius R cover M . Under fairly base. weak assumptions, r R . It is conjectured that both these ratios are much smaller for primitive permutation groups; in particular, The covering radius is thus it is conjectured that a greedy base has size at most ✟✰✆ ✟ times the minimum base size in a primitive R max ✯ M min ✯ S d ✞ x ✄ y 9 ✍ 8 o ✞ 1 x y group. There is a relation between base size b ✟ and ✞ G We now look at covering radius of subsets (and minimal degree µ ✞ G ✟ . Since any base meets the subgroups) of the symmetric group, with the support of any non-identity element, it follows that in Hamming distance. a transitive group G we must have ✟✑✬ µ ✟✮✭ b ✞ G ✞ G n 14 16