Permutation Groups, Error-Correcting Codes and Uncoverings

Transcription

1 Permutation Groups, Error-Correcting Codes and Uncoverings Robert Francis Bailey Queen Mary, University of London Thesis submitted to the University of London for the degree of Doctor of Philosophy 2 nd November 2005

2 2 Abstract We replace the traditional setting for error-correcting codes (i.e. linear codes) with that of permutation groups, with permutations in list form as the codewords. We introduce a decoding algorithm for these codes, which uses the following notion. A base for a permutation group is a sequence of points whose stabiliser is trivial. An uncovering-by-bases (or UBB) is a set of bases such that any combination of error positions is avoided by at least one base in the set. In the case of sharply k-transitive groups, any k-tuple of points forms a base, so a UBB can be formed from the complements of the blocks of a covering design. (In this case, we use the term uncovering.) A large part of the thesis (chapters 2 to 5) is concerned with constructing UBBs for groups which are base-transitive, i.e. which act transitively on their irredundant bases, which were classified by T. Maund. Various combinatorial, algebraic and number-theoretic techniques are employed in this. Other topics include a case study of the Mathieu group M 12, where we investigate ways in which we can improve the performance of our algorithm, and how the group behaves as a code when number of errors received exceeds the correction capability. We also develop an alternative algorithm which works for the group C m S n, and compare its complexity with the original. There is also a chapter about what can be said for more general permutation groups when regarded as codes in this manner, where we formulate a conjecture (the single orbit conjecture ) about UBBs for permutation groups in general, which we prove for some special cases.

3 For my grandparents D. M. J. ( ) H. O. W. J. ( ) 3

4 Contents 1 Introduction The idea Sharply k-transitive groups The algorithm: uncoverings Generalising Why base-transitive groups? Where next? Infinite families of arbitrary rank Introduction Wreath products of regular groups Zero-sum subgroups GL(n,q) AGL(n,q) Comparison of GL(n,q) and AGL(n,q) Base-transitive groups of rank Introduction A general construction

5 5 3.3 A worked example in GAP Base-transitive groups of rank Introduction An uncovering by triples Blow-ups of PGL(2,q) Background material on finite fields Uncoverings-by-bases for GL(3,q) Uncoverings-by-bases for AGL(2,q) Exceptional base-transitive groups The action of A 7 on 15 points Two groups related to A Two Mathieu groups, M 11 and M Further topics: some case studies Introduction Patterns of repeated symbols Optimising the algorithm A measure of optimality Nested uncoverings Calculating the expected duration of the algorithm Some concluding remarks M 12 : words with four errors M 12 : more than four errors

6 6 6.6 C m S n : an alternative algorithm C m S n : more than r errors Other groups: the single orbit conjecture The conjecture Direct and wreath products S m acting on 2-subsets Another family of groups A stronger conjecture Complexity issues Complexity of the uncovering-by-bases algorithm Improvements Complexity of the alternative algorithm Comparison of the two algorithms A Some GAP programs 129 A.1 The decoding algorithm for sharply k-transitive groups A.2 The uncovering-by-bases decoding algorithm A.3 Programs for investigating M A.4 The alternative decoding algorithm A.5 Programs for verifying the single orbit property Bibliography 137

7 Acknowledgements I should begin for thanking my supervisor, Peter Cameron, for all his help and encouragement during the writing of this thesis, being an endless supply of ideas, coping with my inability to understand most of them, and tolerating me spending the first two years turning up to supervisions with blank sheets of paper. This work would not have been possible without the CASE studentship provided by the EPSRC and GCHQ. I should thank Cliff Cocks and Richard Pinch from GCHQ for their interest in this work, and also for not insisting that we worked on Trelllises in Communication Theory (the original project title) which bears pretty much no resemblance whatsoever to the contents of the next 165 pages. I should also thank the members School of Mathematical Sciences at Queen Mary, University of London, for making the place such an enjoyable, and stimulating, place to work. So many people there have shown an interest or helped out that there are too many such people to all be listed individually, but some of them deserve a particular mention: Rosemary Bailey, my second supervisor (especially for the apparently limitless confusion over the rightful ownership of the name Bailey ), Leonard Soicher (who wouldn t let me get away without including chapter 8), Bill Jackson, John Bray, Thomas Prellberg and many more. I m grateful to Sophie Huczynska at the University of St Andrews for taking an interest in what I ve been doing, and in particular for a most useful discussion on finite fields. I m also grateful to Jonathan Dixon and Mike Newman for helping out with that thankless task, proofreading. The past three-and-a-bit years would not have been the same without my fellow occupants of rooms 201, 202 and 203. Again, they have been too numerous to all be listed, but again some are worthy of a particular mention: Jan Grabowski and Jonathan Dixon for all the overly-detailed cricket/latex/london Transportrelated conversations (which frequently would drive others away), John Arhin for patiently putting up with my inability to use GAP, Cheng Yeaw Ku for his flipping 7

8 ACKNOWLEDGEMENTS 8 door, Matt Ollis for the alcohol supply, Becky Thorn for being utterly bonkers (but in a nice way), Debbie Lockett for inadvertantly stalking me, and Rebecca Lodwick for helping to preserve my sanity in room 203, when all around were descending into madness. Of course, I can t possibly forget those who convinced me that being a pure mathematician was what I wanted to be, namely people at the University of Leeds like Dugald Macpherson, John McConnell, Chris Robson, John Truss and (most of all) R. B. J. T. (Reg) Allenby. Finally, I should thank my family for putting up with me whilst doing all of this (rather than making me get a proper job); none of them will understand a word of this, but hopefully will still be impressed by it all the same. While I m at it, I should probably also thank my hairdresser, my therapist,... Robert Bailey London, 2 nd November 2005

9 The results in this thesis, other than those clearly marked, are the unaided work of the author. Signed: Date: 9

10 Chapter 1 Introduction 1.1 The idea This thesis is concerned with error-correcting codes. However, the usual notion of linear codes, which are vector spaces over finite fields, does not feature. Instead, we will use a group of permutations as the code, where the codewords are group elements written in list form, with the usual Hamming distance. We begin by clarifying what we mean by this. In fairly broad generality, we have the following definition. Definition An (error-correcting) code is a set C of strings of symbols, called codewords, chosen from some alphabet. If the strings are all of the same length, then we can make the following definition. Definition Let x = x 1 x 2 x n and y = y 1 y 2 y n be two codewords. Then the Hamming distance is d H (x,y) = #i : x i y i, i.e. the number of co-ordinates in which x and y differ. This then leads naturally to the next definition. Definition The minimum distance of a code C is d(c) = min d H (x,y), x,y C x y i.e. the least value of d H over all pairs of words in C. 10

11 INTRODUCTION 11 This is important for error correction, as if a codeword is transmitted and the received word contains errors, then as long as there are at most r = d(c) 1 2 errors, there will be a unique nearest neighbour in C and we can decode correctly. We call this parameter r the correction capability of C. In our situation, our codewords are strings of length n chosen from the alphabet {1,...,n}, i.e. permutations written in list form. For example, is an element of S 9. Furthermore, in this context the Hamming distance has a useful alternative interpretation. Proposition For g,h S n, d H (g,h) = n Fix(gh 1 ) where Fix(g) denotes the set of fixed points of g. Proof. By applying h 1 to both g and h, d H (g,h) = d H (gh 1,Id) is the number of places gh 1 differs from the identity, which is n Fix(gh 1 ). Thus the minimum distance of a set C of permutations is equal to min n Fix(gh 1 ) = n max Fix(gh 1 ). g,h C g h g,h C g h When the set of permutations forms a group G, this becomes n max Fix(g), g G g 1 which is known to group theorists as the minimum degree of G. There is an analogy here to the theory of linear codes, in that the minimum distance of a linear code is equal to its minimum weight, i.e. the distance from the all-zero codeword, which plays the rôle of the identity permutation in that setting. One way to calculate the minimum distance (i.e. minimum degree) is to use character theory. For any group G acting on a finite set Ω of size n, we define the permutation representation of G to be the map ρ : G GL n (C) g P(g)

12 INTRODUCTION 12 where P(g) i j = { 1 if g(i) = j 0 otherwise. This defines a representation in the usual sense (see, for example, [26] or chapter 2 of [7]). Then the function π : G C g Trace(P(g)) is called the permutation character of G. So π(g) is the number of ones on the diagonal of the matrix P(g), i.e. the number of points in Ω fixed by g. Thus when we consider the elements of G to be permutations written in list form, we have that d H (g,id) = n π(g). So therefore the minimum distance (minimum degree) is n max π(g). g G g 1 For any code, we can define a polynomial which counts the number of codewords at a each distance from a fixed codeword, which is as follows. Definition Let C be a code of length n and minimum distance d. Then for a fixed codeword c C we define the distance enumerator, c (x) = x dh(c,w). w C Clearly, the coefficient of x i gives the number of codewords at distance i from c. In the case where C is a permutation group, we obtain the same polynomial regardless of the choice of c (because each codeword can be mapped to any other by a permutation), so we can take c to be the identity permutation and call the polynomial (x). In the case where C is a linear code, we take c to be the zero vector and obtain the weight enumerator 1. This gives us another analogy with linear codes. Also, we can rephrase the distance enumerator in terms of the permutation character. As is well-known (see for instance [26], proposition 13.5), characters are constant on conjugacy classes. So using the notation above, the distance enumerator becomes (x) = Class(g) x n π(g), g R 1 Usually this is given as a two variable polynomial, W(x,y) = w C x n wt(w) y wt(w), where wt(w) = d H (w,0): see [12], page 121.

13 INTRODUCTION 13 where R is a set of conjugacy class representatives for G, and Class(g) denotes the conjugacy class containing g. Example Let G be the alternating group A 5. Using the character table in the ATLAS [18], page 2, we obtain the polynomial (x) = x x x 5. Note that by evaluating (1), we obtain G. The idea of using permutation groups as codes in this way is not new; in fact it was first decribed in 1974 by Blake [3], while Blake, Cohen and Deza developed the idea further in their 1979 paper [4]. To begin with, we shall concentrate on the same groups as [3], namely sharply k-transitive groups. 1.2 Sharply k-transitive groups Suppose G is a permutation group acting on Ω = {1,...,n} (so G has degree n). Then we say G is sharply k-transitive if for any two ordered k-tuples of distinct elements of Ω, there is a unique group element mapping the first to the second. A sharply 1-transitive group is a regular permutation group, so we restrict ourselves to the case k 2. As Blake points out in [3], the minimum distance of such a group is easy to calculate. Proposition Let G be a sharply k-transitive permutation group of degree n. Then the minimum distance of G is n k + 1. Proof. By definition, only the identity element can fix k points, so the maximum number of fixed points of a non-identity element is k 1. (It is easy to show that such an element exists, for instance mapping (x 1,...,x k 1,x k ) to (x 1,...,x k 1,y).) Hence the minimum distance is n (k 1) = n k + 1. Example The symmetric group S n is both sharply n-transitive and sharply (n 1)-transitive, while the alternating group A n is sharply (n 2)-transitive. Thus the minimum distance of S n is n (n 1) + 1 = 2, and the minimum distance of A n is n (n 2) + 1 = 3. Thus, regarded as a code, the symmetric group is not much use, as its correction capability is = 0. The correction capability of the alternating group is

14 INTRODUCTION However, there are some more interesting examples. In fact, a complete classification of sharply k-transitive groups (for k 2) is available. For k 4, it is a classical result of Jordan from 1873 [28], and the classification was completed (for k = 2 and 3) by Zassenhaus [39, 40] in A sharply k-transitive group must be one of the following. k 6: S k, S k+1 and A k+2 only. k = 5: S 5, S 6, A 7 and the Mathieu group M 12. k = 4: S 4, S 5, A 6 and the Mathieu group M 11. k = 3: The group PGL(2,q) = { τ : x ax + b cx + d } a,b,c,d F q,ad bc 0 acting on the projective line F q { } (which includes S 3 = PGL(2,2), S 4 = PGL(2,3), A5 = PGL(2,4)), plus an additional infinite family as described by Cameron [7], page 16. k = 2: The group AGL(1,F) = {τ : x ax + b a,b F,a 0} where F is a finite near-field, acting on F. In the case where F = F q, we use the notation AGL(1,q). Note that AGL(1,2) = S 2, AGL(1,3) = S 3 and AGL(1,4) = A 4. A near-field is an object which satisfies all of the axioms of a field, with the exception of the commutativity of multiplication and a left distributive law. So any field is a near-field, and there is also an additional infinite family of finite near-fields plus seven exceptional examples. All have prime-power order, and are described by Cameron [7], page 16. A detailed account of the proof of this classification is given by Dixon and Mortimer [20], section 7.6. The Mathieu groups M 11 and M 12 are perhaps the most interesting examples of all, and will be discussed in more detail in later chapters. 1.3 The algorithm: uncoverings Regarded as a code, the correction capability of a sharply k-transitive group G of degree n is r = d 1 2, where d = n k + 1. So our decoding method will assume

15 INTRODUCTION 15 that at most r errors have occurred, i.e. that at least n r symbols are correct. Now, any k-tuple of these correct symbols will uniquely identify the codeword. However, we don t know in which r positions the errors are, so we need a way of choosing a set of k-tuples so that we can be certain that at least one contains no errors. That is, we need a set of k-subsets of {1,...,n} such that any r-subset of X is disjoint from at least one k-set. More formally, we have the following. Definition A set U of k-subsets of Ω = {1,...,n} is called an (n,k,r)- uncovering if it has the property that for any r-subset R Ω, there exists S U such that R S = /0. Provided that k n r, an uncovering will always exist, by taking the set of all k-subsets of Ω. So the problem becomes one of finding the smallest possible uncovering for a given set of parameters. We call an uncovering of least size a minimal (n, k, r)-uncovering. The complementary problem of finding a set of m-subsets of Ω such that every r-subset of Ω (where r < m) is contained in at least one m-set is of interest in design theory. Such a set of m-sets is referred to as an (n,m,r) covering design, and the m-sets are known as blocks. So by taking the the complements of the blocks of an (n,m,r) covering design, one obtains an (n,n m,r)-uncovering. For this reason, we use the term coblocks for the subsets in an uncovering. Consequently, the extensive literature on covering designs is of use to us in finding uncoverings. For instance, there is a large internet database of covering designs with small parameters maintained by Gordon, the La Jolla Covering Repository [22]. Many of the constructions featured in the database are described in the paper of Gordon, Kuperberg and Patashnik [23], while a more general survey can be found in Mills and Mullin [33]. Example For the sharply 3-transitive group PGL(2,7), we have n = 8, k = 3, r = = 2, so need an (8,3,2)-uncovering, as shown below The above example was obtained from an (8,5,2) covering design in Gordon s database [22]. It is small enough for the uncovering property to be verified easily by hand.

16 INTRODUCTION 16 For a given sharply k-transitive group G, once we have a suitable uncovering we can apply the following decoding procedure. Algorithm Suppose we have received the word w = w 1 w 2 w n. Take a k-set S and look at the entries w i for i S. First check that there are no repeated symbols in these positions. If not, then because G is sharply k-transitive, there is a unique permutation mapping each i to w i for i S; that is, there is a unique codeword with entry w i in position i for i S. So we search through the codebook (i.e. the list of elements of G) until we find it. We then check the distance between this codeword and the received word w. If it is within distance r then we conclude that this must be the transmitted word and stop the procedure. If not, we consider the next k-set of co-ordinate positions and repeat the procedure. (If at any stage a k-tuple of entries includes a repeated symbol, then we know that there must be an error here and proceed immediately to the next k-set.) A diagram which describes this algorithm is given in figure 1.1. It has been implemented in the computer system GAP [21], and is given in appendix A.1. START Choose first k tuple Identify corresponding codeword in codebook Choose next k tuple Within distance r of received word? NO YES STOP Figure 1.1: The decoding algorithm Example We continue with the example of PGL(2, 7), which is generated by the permutations ( ) and (1 2 6)(3 4 8) (in disjoint cycle form). Sup-

17 INTRODUCTION 17 pose we transmit the permutation and that the following word is received: g = w = (This has two errors, in positions 1 and 4.) Using the uncovering in example 1.3.2, we first identify the element of PGL(2,7) which maps the 3-tuple (1,2,3) to (4,2,3), which is This is distance 4 from w, so is rejected. Then we look at the next 3-tuple in the uncovering, which is (4,5,6). In w, these positions contain the symbols (6,5,6), so clearly no permutation can exist here. So we move to the next 3-tuple, (2,3,7); these positions contain the symbols (2, 3, 7). Thus we find the element which maps the first to the second, which is This is distance 2 from w, so is accepted, and the algorithm terminates. In this thesis, we find suitable uncoverings for all the sharply k-transitive groups, with the exception of the symmetric group in its natural action on n points. (Since the correction capability of S n is zero, this is a degenerate case.) The alternating group forms a 1-error correcting code, so for each n we require an (n, n 2, 1)-uncovering. Construction We need a set of (n 2)-subsets of {1,...,n} with the property that every point lies outside of one of the subsets. Alternatively, we need an (n,2,1) covering design. If n is even, then we can take 1 2n disjoint pairs, and we are done. If n is odd, then we take 2 1 (n 1) disjoint pairs, then a pair which contains the remaining point and any other point. Then to obtain an (n,n 2,1)- uncovering, we take the complement of each pair. Example For the alternating group A 7, we need a (7,5,1)-uncovering. Using the construction above, we obtain:

18 INTRODUCTION 18 Uncoverings for the remaining sharply k-transitive groups will be given in later chapters. We conclude this section by mentioning that the decoding method we give here is not the already-known method of permutation decoding. This method, which is attributed to F. J. MacWilliams, is a decoding method used for linear codes, involving finding a subset of the automorphism group of the code to move any set of errors out of the information positions. A full description of this method is given in the survey article by Huffman [25] in the Handbook of Coding Theory. While this method is also related to covering designs, it differs from our method in that it involves finding a set of group elements, rather than a set of subsets of the domain of a permutation group. 1.4 Generalising Algorithm 1.3.3, as stated in the previous section, works only for sharply k- transitive groups. However, as the classification of these is quite restricted, then we would like to be able to utilise this algorithm for other groups. The following definition, originally due to Sims [37], is fundamental here. Definition Let G be a group acting on a finite set Ω. A base for G in this action is a sequence of points (x 1,...,x b ) from Ω such that G (x1,...,x b ) = 1, i.e. the pointwise stabiliser is the identity. An irredundant base is a base where G (x1,...,x i,x i+1 ) G (x1,...,x i ) for i = 1,...,b 1. For instance, in a sharply k-transitive group, any sequence of k points forms an irredundant base. The next example is also straightforward. Example Consider the general linear group GL(n, q) acting on the non-zero vectors in F n q. Then a basis for the vector space F n q forms an irredundant base for GL(n,q). We saw in the previous section that in a sharply k-transitive group, the action of a group element on a k-tuple uniquely determines that element. The next proposition shows that bases capture this property. Proposition For any group G, the action of an element g G on a base (x 1,...,x b ) uniquely determines that element; that is, if (x 1,...,x b ) g = (x 1,...,x b ) h, then g = h.

19 INTRODUCTION 19 Proof. Suppose g,h G, (x 1,...,x b ) is a base for G and that x g i = x h i for each i. Then x gh 1 i = x i for each i, that is gh 1 G (x1,...,x b ) = 1. Hence gh 1 = 1, i.e. g = h. So, if a group G is to be used as a code, if the received word contains errors in positions outside those labelled by a base, we can decode successfully. Therefore to apply our algorithm we should replace k-tuple by base. Consequently, we replace uncovering with the following. Definition Suppose G is a group acting on Ω, where Ω = n, with correction capability r. Then an uncovering-by-bases for G is a set of bases for G such that any r-subset of Ω is disjoint from at least one base. The revised algorithm, which uses uncoverings-by-bases rather than uncoverings, is essentially the same as algorithm It has two subtle modifications: first, it allows for the possibility that the bases in the uncovering have different sizes; second, because the existence of a group element agreeing with the received word in the positions labelled by a given base is no longer guaranteed, we include a check for non-existence. As it is of fundamental importance in this thesis, we describe it explicitly below. Algorithm Suppose we have a group G, an uncovering-by-bases U and have received the word w = w 1 w 2 w n. Take the first base B U and look at the positions in w indexed by the entries of B. If they are all distinct, then we check if there exists an element g G which agrees with w in those positions. If so, we then check if g is within distance r from w. If this happens, then we conclude that g was the transmitted word and stop. Otherwise, we move to the next base in U and repeat the procedure. This algorithm has been implemented in GAP and is given in appendix A.2. The procedure used to determine the element agreeing with the images of a base is a standard technique in computational group theory, and is described in section 8.1, where we determine the complexity of algorithm Clearly, in the case where G is sharply k-transitive, an uncovering-by-bases is just an (n, k, r)-uncovering, as defined above. Unlike that situation, in general it is not immediately obvious that for an arbitrary group an uncovering-by-bases should exist. However, we are saved by the next result. Proposition For any finite group G acting on a set Ω with Ω = n, there always exists an uncovering-by-bases.

20 INTRODUCTION 20 Proof. Let d be the minimum distance of G, so r = d 1 2. We show that for an arbitrary r-subset of Ω, there exists a base for G disjoint from it, arguing by contradiction. Suppose there exists an r-subset R Ω that meets every base for G. Then the pointwise stabiliser of R = Ω\R is non-trivial, as R does not contain a base. Therefore there exists a non-identity element g that fixes R pointwise, so Fix(g) R = n r. But the maximum number of fixed points of a non-identity element is n d < n r, giving a contradiction. We remark that the definition of uncovering-by-bases, and indeed the proof of proposition 1.4.6, is vacuous in the case r = 0. Although groups with zero correction capability are, of course, useless as error-correcting codes, an uncovering-bybases for such a group consists of a single base only. Proposition is, of course, only an existence result: it gives us no idea as to how to set about finding them. A sensible starting point would be groups whose base structure is known. 1.5 Why base-transitive groups? The following definition is due to Cameron and Fon-Der-Flaass [11]. Definition A permutation group in a given action is called an IBIS group if all irredundant bases have the same size. The acronym stands for irredundant bases of invariant size. Examples of IBIS groups include the sharply k-transitive groups and general linear groups we mentioned above. The easiest example of a group that is not an IBIS group is the symmetric group acting on 2-subsets; this will be decribed in chapter 7. In their 1995 paper [11], Cameron and Fon-Der-Flaass proved the following theorem. Theorem The following statements are equivalent: 1. G is an IBIS group; 2. the irredundant bases of G are preserved by re-ordering; 3. the irredundant bases of G form the bases of a matroid.

21 INTRODUCTION 21 A matroid is a set together with a family of subsets, called independent sets, which satisfy certain axioms generalising the notion of linear independence in a vector space. (A full treatment is given in Oxley s book [35].) The maximal independent sets are called bases and the size of a base is the rank of the matroid. (Bases must all have the same size: see [35], page 16.) If G is an IBIS group, we call the size of an irredundant base the rank of G, as this is the rank of the corresponding matroid. It is the second property in theorem that is of use to us here: if we know that all the irredundant bases of G have the same size, then we can order them in any way we choose, and so can regard a base as a set rather than a sequence. This should hopefully ease the construction of an uncovering-by-bases. Unfortunately, a complete classification of IBIS groups seems unlikely. Thanks to a construction due to Cameron [8], any linear code gives rise to an IBIS group, so a classification of IBIS groups would include a classification of linear codes. However, in a more special situation, a complete classification is available. Definition If a group acts transitively on its irredundant bases, we say it is a base-transitive group. Obviously, sharply k-transitive groups are base-transitive, and base-transitive groups are IBIS groups. In the literature, the rather ambiguous name of geometric groups is often used for base-transitive groups. This name is due to Cameron and Deza [9]; it arises from the fact that the associated matroids are permutation geometries. Base-transitive groups of rank 1 are regular permutation groups. However, for rank k 2, a complete classification of base-transitive groups is known. Part of the classification involves determining the type of each basetransitive group, which is defined as follows. Definition Let G be a base-transitive group of degree n and rank k, and let (x 1,...,x k ) be an irredundant base for G. Then the type of G is the pair ({l 0,l 1,...,l k 1 },n), where l 0 is the number of fixed points of G, and l i is the number of fixed points of G (x1,...,x i ) for 1 i k 1. Since l k 1 is the maximum number of fixed points of a non-identity element, we can therefore obtain the correction capability of G from its type. Essentially a base-transitive group falls into one of the following cases: generic examples for arbitrary rank;

22 INTRODUCTION 22 infinite families for ranks 2 and 3; sporadic examples of rank 5. The generic examples are as follows: (i) S n, which has rank n 1 and type ({0,1,...,n 2},n); (ii) A n, rank n 2, type ({0,1,...,n 3},n); (iii) G = H S n, where H is a regular group of degree m, so G has rank n and type ({0,m,...,(n 1)m},nm); (iv) the semidirect product G = X S n, where X = {(a 1,a 2,...,a n ) A n a 1 + a a n = 0} (A is an abelian, regular group of degree m, written additively), so G has rank n 1 and type ({0,m,...,(n 2)m},nm); (v) GL(n,q), rank n, type ({0,q 1,q 2 1,...,q n 1 1},q n 1) (i.e. the general linear group); (vi) the stabiliser of d independent vectors in GL(n,q), where 0 < d < n, which has rank n d and type ({0,(q 1)q d,...,(q n 1 )q d },(q n 1 )q d ); (vii) AGL(n,q), rank n + 1, type ({0,q,q 2,...,q n 1 },q n ) (i.e. the affine general linear group); (viii) the group V H, where V is the additive group of F n q and H is the group in (vi), this has rank n d + 1 and type ({0,q d+1,...,q n 1 },q n ). Zil ber [41] showed that for rank at least 7, there are only generic examples. The classification for ranks 2 to 6 is due to Maund [31]: we summarise her result below. In addition to the appropriate generic examples, we have the following groups. Rank 2 the sharply 2-transitive groups, which have type ({0,1},n) (where n is a prime power); C (q 1)/2 PSL(2,q) for q 3 mod 4, of type ({0, q 1 2 }, (q2 1) 2 );

23 INTRODUCTION 23 C q 1 Sz(q), type ({0,q 1},(q 1)(q 2 +1)) (where q is an odd power of 2); PSL(3,2), type ({0,2},14); PSL(3,3), type ({0,6},78). In the last four cases, the base-transitive action is on the set of right cosets of particular subgroups. Rank 3 PGL(2,q) and the other sharply 3-transitive groups, type ({0,1,2},q + 1); blow-ups of PGL(2,q), type ({0,q d,2q d },q d (q + 1)) (see below); A 7 acting on F 4 2 \ {0}, type ({0,1,3},15); V H, where V is the additive group of F 4 2 and H is the point-stabiliser of A 7 in the above action, type ({0,2,4},16). Ranks 4, 5 and 6 V A 7 (where V is as above), rank 4, type ({0,1,2,4},16); M 11, rank 4, type ({0,1,2,3},11); M 12, rank 5, type ({0,1,2,3,4},12). Unlike Zil ber, Maund s proof uses the classification of finite simple groups. It involves first classifying the base-transitive groups of rank 2 (by using the classification of 2-transitive groups), and then using induction to analyse the higher ranks. Unfortunately, neither Maund nor Zil ber s work has ever been published in a particularly accessible form: Cameron [6] contains a brief survey. One important notion in the study of base-transitive groups is that of a blowup. The construction was introduced by Cameron, Deza and Frankl [10] for permutation geometries. In the language of groups, it is as follows. Definition Let G be a base-transitive group of rank k and type (L,n). Then a blow-up of G is a base-transitive group Ĝ of rank k and type (ml,mn) (where ml = {ml l L}), such that:

24 INTRODUCTION 24 (i) Ĝ has a normal subgroup N which has n orbits of size m, (ii) Ĝ/N = G, (iii) Ĝ induces the given action of G on the N-orbits. From the third property, it follows that a base for Ĝ consists of k points chosen from k independent N-orbits, i.e. from orbits which form a base for the action of G induced on them. The symmetric group S n is something of an anomaly here, as although it has rank n 1 and type ({0,1,...,n 2},n), when it comes to construct blow-ups, it behaves as if it also has an action of rank n and type ({0,1,...,n 1},n). (This is because it is both sharply (n 1)-transitive and sharply n-transitive.) We see this in the following example. Example Consider the entry (iii) in the list of generic base-transitive groups. Here we have G = S n and Ĝ = H S n, where H is a regular group of order m. Since H S n means H n S n, we take K = H n (the direct product of n copies of H). So K has n orbits of size m, Ĝ/K = S n, and Ĝ induces the usual action of S n on these orbits. In the list of generic examples, entries (iv), (vi) and (vii) are all blow-ups, of S n, GL(n d,q) and AGL(n d,q) respectively. There is also a family of blow-ups of PGL(2,q), but these are quite complicated to describe. Most of this thesis (chapters 2, 3, 4 and 5) is devoted to the construction of uncoverings-by-bases for base-transitive groups, employing a variety of methods. 1.6 Where next? Although most of this thesis is devoted to base-transitive groups, and constructing uncoverings-by-bases for them, we also consider some other topics. In chapter 7, we consider permutation groups in more generality. In particular, we formulate a conjecture (the single orbit conjecture ) which asserts that any finite permutation permutation group G not only has an uncovering-by-bases, but one which is contained in an orbit of G on irredundant bases. While we do not prove this conjecture (which is probably quite hard in fullest generality), we prove

25 INTRODUCTION 25 some reduction theorems and also consider some special cases. One of these special cases (the action of S m on 2-subsets of {1,...,m}) uses techniques borrowed from the theory of graph decompositions. One of the questions left open at the end of this thesis is to prove this conjecture, and some stronger versions described in section 7.5. In chapter 6 we investigate some different aspects of this theory of using permutation groups as codes. For instance, we investigate methods of improving the decoding algorithm, usually with the aid of the Mathieu group M 12 as a case study. Section 6.2 simply investigates how patterns of repeated symbols can be used to increase the efficiency of the decoding algorithm. Section 6.3 considers optimality in more detail, discussing exactly what optimality means here, ways to measure it, and what steps could be taken to improve it. This in turn gives rise to some interesting questions about covering designs in general. While it is clear that an (n,m,r) covering design is also an (n,m,s) covering design for all s {1,...,r}, an optimal (n,m,r) design is not an optimal (n,m,r 1) design, and may not even contain one. This then leads to different notions of nesting covering designs. There is plenty of scope for future work in this area also. Also in chapter 6, we also develop an alternative algorithm which works in a special case (the generic family of groups C m S n ). We study how this algorithm can be utilised to correct more than r errors (where r is the correction capability). We also investigate such a capability for M 12 with our original algorithm. Investigating this for other groups is another possibility for future research. In chapter 8, we determine the complexity of algorithm and explain why the single orbit conjecture from chapter 7 is beneficial with regard to this. We also determine the complexity of the alternative algorithm, and use this to compare the performance of the two algorithms.

26 Chapter 2 Infinite families of arbitrary rank 2.1 Introduction In this chapter we consider some of the generic base-transitive groups discussed in the previous chapter in more detail. We determine their parameters as errorcorrecting codes and construct uncoverings-by-bases for them. We have already seen (in construction 1.3.5) how to find an uncovering-by-bases for the alternating group A n. Entries (iii) and (iv) on the list (i.e. blow-ups of S n ) are dealt with completely in the following two sections. Finally, we consider the general linear and affine general linear groups; however, as will be seen, these are only partial results, as the uncoverings we construct do not utilise the full correction capabilities of these groups. 2.2 Wreath products of regular groups These are the third entry on the list of generic base-transitive groups, and have already been mentioned in example 1.5.6, as an example of a blow-up. Let H be any regular permutation group of order m. H could be any group of order m acting on itself by right-multiplication, for instance the cyclic group of order m. Then consider the permutation group G = H S n, that is the wreath product of n copies of H by the symmetric group S n. The domain of G, which we call Ω, consists of n copies of a set of size m, which we think of as n columns each of m rows, as depicted below

27 INFINITE FAMILIES OF ARBITRARY RANK 27 G acts on the columns as S n, and the kernel of this action (i.e. the group fixing all the columns) is H n. To apply our decoding algorithm to use G as an error-correcting code, we need to know the structure of the bases of G, and also the minimum degree of G. These questions are answered by the two results below. First, however, we need the following definition. Definition A transversal of Ω is a set of n points in Ω, formed of a single point from each of the n columns of Ω. For example, the shaded points in the diagram below form a transversal..... Proposition A transversal of Ω forms a base for G. Proof. Let x = (x 1,...,x n ) be a transversal. To show that x is a base, we have to show that the stabiliser of the columns containing points from x is trivial, and then that x is a base for H n. The first part is clear, since as every column contains a point from x then the stabiliser of these columns in S n must be the identity. Now suppose that h H n lies in the stabiliser of x. As the action of H n is regular on each column, if one point in a column is fixed by an element, then all points in that column are fixed. It then follows that h must fix all points in all columns, so therefore h must be the identity element of H n. Hence x is a base for H n, and so is also a base for G = H S n. It is easy to see that G acts transitively on the set of all transversals; to map one to another merely requires applying the appropriate cyclic permutation in each column. It is also quite easy to determine the minimum degree of G. Proposition The maximum number of fixed points of a non-identity element of G is (n 1)m, and so the minimum degree of G is m.

28 INFINITE FAMILIES OF ARBITRARY RANK 28 Proof. As mentioned in the proof of Proposition 2.2.2, if a point in a column of Ω is fixed, then all points in that column must be fixed, so therefore fixed points occur in multiples of m. Since the action is faithful, no non-identity element of G can fix all nm points of Ω, so the most that can be fixed is (n 1)m. However, we need to show that such an element exists. Consider an element of G where the element from S n permuting the columns is trivial (i.e. all columns stay where they are), and where all columns except one are fixed. This element has the required number of fixed points. Hence the minimum degree of G is nm (n 1)m = m. Immediately, we have the usual corollary about the number of errors we can correct. Corollary When used as an error-correcting code, the correction capability of G is r = m 1 2. Thus it remains to construct an uncovering-by-bases for this group. However, this is surprisingly straightforward. Recall from Proposition that a base for G consists of a transversal of Ω. Theorem A set of r + 1 disjoint transversals of Ω forms an uncovering-bybases for G. Proof. For simplicity, we take the first r+1 rows of Ω to be our disjoint transversals. Now, if there are r errors, these could be spread across many columns, but can appear in at most r different rows. Thus at least one of the r + 1 transversals must avoid the error positions. A similar technique is required for the related family of groups described in the next section. 2.3 Zero-sum subgroups In this section, we consider entry (iv) on the list of generic groups. These are also examples of blow-ups of S n, in its base-transitive action of rank n 1 and type ({0,1,...,n 2},n). Let A be a finite abelian group of order m, written additively. Define X A n as follows: X = {(a 1,a 2,...,a n ) a 1 + a a n = 0}.

29 INFINITE FAMILIES OF ARBITRARY RANK 29 The group we are interested in is G = X S n, where the action of S n on X is by permuting the copies of A. We refer to G as the zero-sum subgroup of A S n. As a permutation group, acting on the copies of A, this has degree mn, where an element (a 1,a 2,...,a n ) X acts by componentwise addition. Proposition A set n 1 points, each chosen from a different copy of A, forms a base for G. We call such a base a partial transversal of A n. Proof. We use the same pattern of argument as we did with proposition Let a = (a 1,...,a n 1 ) be a partial transversal. First we consider the action of S n on A n. As we have n 1 points chosen from distinct copies of A, if an element of S n fixes these copies it must also fix the remaining copy. Hence the stabiliser in S n of the n 1 copies must be trivial. Now we show that a is a base for X. Choose some g X which stabilises a. Since g X, we have that g = (g 1,g 2,...,g n ). By definition, we have (a 1,a 2,...,a n 1 ) g = (a 1 + g 1,a 2 + g 2,...,a n 1 + g n 1 ) = (a 1,a 2,...,a n 1 ) (since g lies in the stabiliser of (a 1,a 2,...,a n 1 )). Hence g 1 = g 2 = = g n 1 = 0 A. But we then have n i=1 g i = n 1 g i + g n i=1 = 0 A + g n = g n (by the definition of X), so therefore g n = 0 A also. Consequently, we have g = (0 A,0 A,...,0 A ). Therefore the stabiliser of a in X is trivial, so a forms a base for X and thus also does for G = X S n. Furthermore, we have this next result. Theorem G acts transitively on the set of all partial transversals, so therefore is base-transitive. Proof. Given two partial transversals a = (a 1,...,a n 1 ) and b = (b 1,...,b n 1 ), we need to demonstrate the existence of some g G with a g = b.

30 INFINITE FAMILIES OF ARBITRARY RANK 30 Since S n acts transitively on the labels, we can assume both a and b are both inside the first n 1 copies of A. Thus it also suffices to find some h X with a h = b. Suppose h = (h 1,...,h n ), where n i=1 h i = 0. Then we have a h = (a 1 + h 1,a 2 + h 2,...,a n 1 + h n 1 ) = b = (b 1,b 2,...,b n 1 ) by choosing h 1 = b 1 a 1,...,h n 1 = b n 1 a n 1. This subsequently determines h n, as we require the sum to be zero. Thus we have constructed a suitable h, and so G acts transitively on partial transversals. As usual, we also need to know the minimum degree of G. Proposition The maximum number of fixed points of a non-identity element of G is (n 2)m, and so the minimum degree of G is 2m. Proof. Since A acts regularly, fixed points must occur in multiples of m. However, unlike the group in section 2, we can t have (n 1)m fixed points here. Recall from the proof of Proposition that G has an S n -action and an X-action. In the S n -action, a non-identity element of S n has at most n 2 fixed points, so if there were an element of G with (n 1)m fixed points, the S n -action would have to be trivial. So we consider the X-action. Recall that X A n. If g = (g 1,g 2,...,g n ) were to fix (n 1)m points, it would have to look like (g 1,0,...,0), where g 1 0, which clearly does not belong to X. Hence it is impossible for an element of G to have (n 1)m fixed points. Now we must show that there do exist elements of G with (n 2)m fixed points. Again, suppose the S n -action is trivial. There exist elements of X with (n 2)m fixed points, for example (g 1, g 1,0,...,0), where g 1 0. Hence the minimum degree of G is nm (n 2)m = 2m. Once more, we have the usual corollary about the correction capability of G. Corollary When used as an error-correcting code, G has correction capability m 1. Proof. We have r = 2m 1 2 = m 1. Finally, to enable us to use our decoding algorithm with G, we will need an uncovering-by-bases.

31 INFINITE FAMILIES OF ARBITRARY RANK 31 Theorem A set of m disjoint partial transversals forms an uncovering-bybases for G. Proof. The proof is analogous to that of Theorem For simplicity, we take the first n 1 entries of each of the m rows of Ω as our partial transversals. Now, if there are r = m 1 errors, then these can be spread across many columns, but can occur in at most m 1 rows. Hence there will be at least one row which contains no error positions, so there will be a partial transversal which avoids them all. 2.4 GL(n, q) We consider the natural action of GL(n,q) on the non-zero vectors of the space F n q. Since GL(1,q) is just the multiplicative group of F q, which is cyclic, we consider the cases where n 2. The degree of this action is q n 1, and a basis for the vector space forms a base for this group in this action. Calculating the minimum degree is equivalent to counting the maximum number of fixed points, so we have the following lemma. Lemma Suppose M GL(n,q) fixes v 1,...,v k. Then M fixes the space that they span, v 1,...,v k. Proof. Clearly Fix(M) is a subspace of F n q, as for x,y Fix(M) and λ F q, we have M(x+y) = Mx+My = x+y and M(λx) = λmx = λx. Now by assumption, we have Mv 1 = v 1,...,Mv k = v k. Choose some w v 1,...,v k. Then so therefore w = k i=1 ( ) k Mw = M λ i v i = i=1 λ i v i, where λ i F q, k i=1 λ i Mv i = k i=1 λ i v i = w. This immediately leads to this result.

32 INFINITE FAMILIES OF ARBITRARY RANK 32 Proposition The minimum degree of GL(n,q) is q n q n 1. Proof. By lemma 2.4.1, the most that can be fixed by a non-identity element of GL(n,q) is a subspace of F n q of codimension 1, which contains q n 1 1 non-zero vectors. Thus max Fix(g) = g GL(n,q) qn 1 1, so the minimum degree is (q n 1) (q n 1 1) = q n q n 1. Again, the correction capability is an immediate corollary. Corollary As an error-correcting code, the correction capability of GL(n, q) is r = 1 2 (qn q n 1 ) 1. Proof. By the standard result that r = d 1 2 (where d is the minimum degree) and observing that q n q n 1 is always even for n 2, the result follows. However, finding infinite families of (q n 1,n, 1 2 (qn q n 1 ) 1) uncoveringsby-bases for arbitrary n and q has proved extremely difficult. The only general construction we have for arbitrary n only yields an uncovering-by-bases utilising the full correction capability for n = 2. (This is actually superfluous, as an alternative construction is given in the next chapter which works for any rank 2 group.) For the case n = 3, a construction is given in chapter 4. What we present here is a method for finding a (q n 1,n, qn 1 n 1) uncovering-by-bases. First we show how to construct a general (v,k, k v 1)-uncovering. Construction Suppose we have a set X of size v. Put t = k v 1. Now, the largest number of disjoint k-subsets of X is k v = t +1. These t +1 disjoint k-sets must uncover all t-subsets of X, as it is not possible for for a t-set to intersect with t + 1 sets. So we have a (v,k, k v 1)-uncovering. So, to find the uncovering-by-bases we require, we must find a set of qn 1 n disjoint bases for the vector space F n q. Recall that the extension field F q n can be regarded as a vector space of dimension n over F q. (The techniques we use here

33 INFINITE FAMILIES OF ARBITRARY RANK 33 and the terminology used is explained in more detail in chapter 4.) Using fieldtheoretic notation, we have F q n = F q (α), where α is the root of some irreducible polynomial in F q [x] of degree n. Then F q (α) has as a basis over F q. B = { 1,α,α 2,...,α n 1} Now we can suppose that α is a primitive element of F q n. That is, α is a generator of the multiplicative group of F q n, which is cyclic, so each non-zero element of F q n can be written as a distinct power of α. In particular, we observe that any translate of B (i.e. the set obtained from B by multiplying each element by some given power of α) is also a basis of F q n = F q (α). We obtain the qn 1 n disjoint bases of Fn q by first obtaining disjoint bases for F q (α), then using the isomorphism from F q (α) to F n q. The disjoint bases for F q (α) are obtained by taking translates by powers of α n of the standard basis B, e.g. B = { 1,α,...,α n 1} α n B = { α n,α n+1,...,α 2n 1} α 2n B = { α 2n,α 2n+1,...,α 3n 1} and so on. We can find qn 1 n disjoint translates of B in this way (including B itself). Now, each of these powers of α can be expressed in terms of elements of B. Reading off the coefficients in order gives us an element of F n q. For each basis of F q (α), these n coefficient vectors then give us a basis for F n q. Consequently, we now have qn 1 n disjoint bases for Fn q. Example We consider the case q = 3, n = 3. Here we have F 27 = F 3 (α),