Brian Mortimer

The primitive permutation groups of degree less than 1000

Our object is to describe all of the-primitive permutation groups of degree less than 1000 together with some of their significant properties. We think that such a list is of interest in illustrating in concrete form the kinds of primitive groups which arise, in suggesting conjectures about primitive groups, and in settling small exceptional cases which often occur in proofs of theorems about permutation groups. The range that we consider is large enough to allow examples of most of the types of primitive group to appear. Earlier lists (of varying completeness and accuracy) of primitive groups of degree d have been published by: C. Jordan (1872) [21] for d ≤ 17, by W. Burnside (1897) [5] for d ≤ 8, by Manning (1929) [34–38] for d ≤ 15, by C. C. Sims (1970) [45] for d ≤ 20, and by B. A. Pogorelev (1980) [42] for d ≤ 50. Unpublished lists have also been prepared by C. C. Sims for d ≤ 50 and by Mizutani[41] for d ≤ 48. Using the classification of finite simple groups which was completed in 1981 we have been able to extend the list much further. Our task has been greatly simplified by the detailed information about many finite simple groups which is available in the Atlas of Finite Groups which we will refer to as the Atlas [8].

The Design of a High-Performance Error-Correcting Coding Scheme for the Canadian Broadcast Telidon System Based on Reed-Solomon Codes

Error correction can greatly improve the performance and extend the range of broadcast teletext systems. In this paper, the requirements for an error-correcting scheme for broadcast teletext in North America (NABTS) are set down. An error-correction scheme which meets all these requirements is then described. The simplest case employs the one parity bit in each 8 bit byte and no suffix of parity check bits at the end of each data block. The next level also uses a single byte of parity check bits at the end of each data block. Adding a second byte of parity checks at the end of each data block results in a Reed-Solomon code, called the C code, for each data block. Adding one data block of parity checks after h - 1 data blocks results in a set of h data packets being encoded into a bundle, in which vertical C codes provide powerful interleaving. In a final alternative, two data blocks hold the check bytes for the vertical codewords, and the most powerful coding scheme, the double bundle code, results. The detailed mathematical definitions of the various codes are referred to or described, formulas for performance calculations are referred to, and performance curves are presented for the AWGN channel as well as for measured field data. These performance curves are discussed and compared to the performance of a difference set cyclic code, originally designed for the Japanese teletext system, which corrects any 8 bits in error in a packet.

Performance of a powerful error-correcting and -detecting coding scheme for the North American Basic Teletext System (NABTS) for random independent errors: methods, equations, calculations and results

For a powerful layered, upward- and downward-compatible error-correcting and error-detecting scheme for NABTS, various bit error rate (BER) related performance measures are derived and calculated for random independent errors. The methods, equations, calculations and results are given for the least powerful one-byte suffix codes, for the two-byte suffix code, called code C, and for the double and single bundle codes formed by using code C for each data block (i.e. horizontally) and also vertically, thus forming a product code, for a specified, but variable, number of data blocks. Performance bounds and equations for probabilities of correct decoding of error and of decoding failure are given. The weight enumerators for a number of one-byte suffix codes are calculated, and those of weight four are classified into types depending on the number of ones occurring in a byte, and in other arrangements. Performance analyses and comparisons with a code for Japanese teletext are included. Analyses used in computer simulation studies are described. >

Bounds for byte-oriented error-correcting codes with application to teletext systems

Codes formed bytes of known parity with one byte of check bits are considered. A lower bound on the number of weight four codewords is obtained. An application to the choice of error-correcting code for the North American Broadcast Teletext Specification is discussed.

The Basic Ideas

A cube is highly symmetric: there are many ways to rotate or reflect it so that it moves onto itself. A cube with labeled vertices is shown in Fig. 1.1. For example, we can rotate it by 90° about an axis through the centres of opposite faces, or reflect it in the plane through a pair of opposite edges. Each of these “ symmetries” of the cube permutes the eight vertices in a particular way, and knowing what happens to the vertices is enough to tell us what the whole motion is. The symmetries of the cube thus correspond to a subgroup of permutations of the set of vertices, and this group, an algebraic object, records information about the geometric symmetries.

Correspondence: A correction to a recent analysis of a product code

In this paper we correct an error in a recent analysis (Reference 1) of a single error correcting product code in the presence of white gaussian noise using a variety of decoding strategies.

Methods and formulas for computer simulation studies of the performance of error-correcting codes for the North American Basic Teletext System (NABTS)

The rationale, methods and formulas used in computer simulation studies of the error performance of the various constituent error-correcting codes designed for the North American Basic Teletext System are described for the case when the input error sequences have an arbitrary distribution. Expressions for the overall packet rejection rate, R, and the number of picture-description instruction (PDI) errors are derived using a classification of the undetected errors in the extended Hamming

Bounds on Orders of Permutation Groups

The theme of the present chapter is use of combinatorial methods to bound the order of various classes of subgroups of the finite symmetric groups. Typically we find that, excluding Anand Sn themselves, the larger subgroups of S n are either intransitive or imprimitive (Theorem 5.2B).

The Structure of the Symmetric Groups

The present chapter studies the symmetric groups with particular emphasis on the infinite case. Of course we know that every group is isomorphic to a subgroup of some symmetric group (for example, via its regular permutation representation), so one might suppose that it is not possible to say much useful about the symmetric groups unless we know a great deal about groups in general. However this is not true. There are certain facts which are available without a detailed knowledge of all of the subgroups, much in the same way that we have useful results about the set of real numbers without knowing detailed facts about individual real numbers.

Examples and Applications of Infinite Permutation Groups

The object of this chapter is to give a selection of examples of infinite permutation groups, and a few of the ways in which permutation groups can be used in a more general context. For example, we give a criterion of Serre for a group to be free which leads to a classic theorem on free groups due to J. Nielson and O. Schreier, and give a construction due to N. D. Gupta and S. Sidki of an infinite p-group which is finitely generated. What makes these constructions manageable is that the underlying set on which the groups act have certain relational structures. The most symmetric of these structures (the ones with the largest automorphism groups) are the homogeneous structures; of these the countable universal graph is an especially interesting and well-studied example.