The covering number of M24M_{24}

A finite cover $\mathcal{C}$ of a group $G$ is a finite collection of proper subgroups of $G$ such that $G$ is equal to the union of all of the members of $\mathcal{C}$. Such a cover is called {\em minimal} if it has the smallest cardinality among all finite covers of $G$. The covering number of $G$, denoted by $\sigma(G)$, is the number of subgroups in a minimal cover of $G$. In this paper the covering number of the Mathieu group $M_{24}$ is shown to be 3336

Some perpendicular arrays for arbitrarily large t

AbstractWe show that perpendicular arrays exist for arbitrarily large t and with λ = 1. In particular, if d devides (t+1) then there is a PA1(t, t+1, t+(f(t+1)d)). If υ ≡ 1 or 2 (mod 3) then there is a PAλ(3, 4, υ) for any λ. If 3 divides λ then there is a PAλ(3, 4, υ) for any v. If n⩾2 there is a PA1(4, 5, 2n+1). Using recursive constructions we exhibit several infinite families of perpendicular arrays with t⩾3 and relatively small λ. We finally discuss methods of constructing perpendicular arrays based on automorphism groups. These methods allow the construction of PA's with (k−t)>1

A Public Key Cryptosystem Based on Non-abelian Finite Groups

Abstract We present a new approach to designing public-key cryptosystems, based on covers and logarithmic signatures of nonabelian finite groups. Initially, we describe a generic version of the system for a large class of groups. We then propose a class of 2-groups and argue heuristically about the system's security. The system is scalable, and the proposed underlying group, represented as a matrix group, affords significant space and time efficiency

Simple method for enhancing the performance of lossy plus lossless image compression schemes

Lossy plus lossless techniques for image compression split an image into a low-bit-rate lossy representation and a residual that represents the difference between this low-rate lossy image and the original. Conventional schemes encode the lossy image and its lossless residual in an independent manner. We show that making use of the lossy image to encode the residual can lead to significant savings in bit rate. Further, the complexity increase to attain these savings is minimal. The savings are achieved by capturing the inherent structure of the image in the form of a noncausal prediction model that we call a prediction tree. This prediction model is then used to transmit the lossless residual. Simulation results show that a reduction of 0.5 to 1.0 bit/pixel can be achieved in bit rates compared to the conventional approach of independently encoding the residual

Lexical Permutation Sorting Algorithm

In this paper, we describe the Lexical Permutation Sorting Algorithm (LPSA), its theoretical basis, and deligneate its relationship to the Block Sorting Lossless Data Compression Algorithm (BSLDCA) described by Burrows and Wheeler [1]. In particular we describe how BSLDCA can be reduced to LPSA and point out its possible applications. 1 Introduction Recently, M. Burrows and D. J. Wheeler [1] introduced a new algorithm, which they call the Block Sorting Lossless Data Compression Algorithm (BSLDCA) for lossless data compression. When applied to text or image data their algorithm is claimed to achieve better compression rates than Gzip, Comp-2 and Ziv-Lempel's LZW algorithm, which is widely known to unix users as, compress. In this paper, we define the Lexical Permutation Sorting Algorithm (LPSA), and show that, BSLDCA is reducible to LPSA. The advantage of going to LPSA is that we now have a clear understanding of its theoretical foundations, and have some optimization choices not avai..

On the Number of Mutually Disjoint Cyclic Designs and Large Sets of Designs

Let N(t; v; k; ) be the maximum possible number of mutually disjoint cyclic t- (v; k; ) designs. In this paper we find the number N(t; v; k; ) for two sets of parameters, and give bounds for other sets of parameters. We also discuss optimization techniques for finding (mutually disjoint) cyclic designs and prove an extension theorem for large sets of designs. The results of this study show the existence of two new large sets of designs. keywords: combinatorial designs, t\Gammadesigns, large sets 1 Introduction Let D = fB 1 ; B 2 ; :::; B b g be a finite family of k-subsets (called blocks) of a point set X = X v = f1; 2; :::; vg. Then, (X; D) is a t-(v; k; ) design if every t-subset of X is contained in exactly blocks of D. Frequently, the point set X is implicit and we simply think of the design as the collection of blocks D. The set of all k-subsets of X v will be denoted here by X (k) v . We will use X (k) instead of X (k) v whenever the value of v is clear from the conte..