Space Forms and Group Resolutions: the tetrahedral family

The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of the integers over G results from the associated G-equivariant cellularization of S. This technique is applied to the generalized binary tetrahedral group family; the homology groups, the cohomology rings and the Reidemeister torsions of the related spherical space forms are determined.Comment: 37 pages, 5 figures. A proof of the minimal rank property for the resolution has been adde

Error-Correcting Codes Associated With Generalized Hadamard Matrices Over Groups

Classical Hadamard matrices are orthogonal matrices whose elements are ±1. It is well-known that error correcting codes having large minimum distance between codewords can be associated with these Hadamard matrices. Indeed, the success of early Mars deep-space probes was strongly dependent upon this communication technology. The concept of Hadamard matrices with elements drawn from an Abelian group is a natural generalization of the concept. For the case in which the dimension of the matrix is q and the group consists of the p-th roots of unity, these generalized Hadamard matrices are called “Butson Hadamard Matrices BH(p, q)”, first discovered by A. T. Butson [6]. In this dissertation it is shown that an error correcting code whose codewords consist of real numbers in finite Galois field Gf( p) can be associated in a simple way with each Butson Hadamard matrix BH(p, q), where p \u3e 0 is a prime number. Distance properties of such codes are studied, as well as conditions for the existence of linear codes, for which standard decoding techniques are available. In the search for cyclic linear generalized Hadamard codes, the concept of an M-invariant infinite sequence whose elements are integers in a finite field is introduced. Such sequences are periodic of least period, T, and have the interesting property, that arbitrary identical rearrangements of the elements in each period yields a periodic sequence with the same least period. A theorem characterizing such M-invariant sequences leads to discovery of a simple and efficient polynomial method for constructing generalized Hadamard matrices whose core is a linear cyclic matrix and whose row vectors constitute a linear cyclic error correcting code. In addition, the problem is considered of determining parameter sequences {tn} for which the corresponding potential generalized Hadamard matrices BH(p, ptn) do not exist. By analyzing quadratic Diophantine equations, new methods for constructing such parameter sequences are obtained. These results show the rich number theoretic complexity of the existence question for generalized Hadamard matrices

Group divisible designs, GBRDSDS and generalized weighing matrices

We give new constructions for regular group divisible designs, pairwise balanced designs, generalized Bhaskar Rao supplementary difference sets and generalized weighing matrices. In particular if p is a prime power and q divides p - 1 we show the following exist; (i) GDD (2(p2+p+1), 2(p2+p+1), rp2,2p2, λ1 = p2λ, λ2 = (p2-p)r, m=p2+p+1,n=2), r_+1,2; (ii) GDD(q(p+1), q(p+1), p(q-1), p(q-1),λ1=(q-1)(q-2), λ2=(p-1)(q-1)2/q,m=q,n=p+1); (iii) PBD(21,10;K),K={3,6,7} and PDB(78,38;K), K={6,9,45}; (iv) GW(vk,k2;EA(k)) whenever a (v,k,λ)-difference set exists and k is a prime power; (v) PBIBD(vk2,vk2,k2,k2;λ1=0,λ2=λ,λ3=k) whenever a (v,k,λ)-difference set exists and k is a prime power; (vi) we give a GW(21;9;Z3)

The Niederreiter cryptosystem and Quasi-Cyclic codes

McEliece and Niederreiter cryptosystems are robust and versatile cryptosystems. These cryptosystems work with any linear error-correcting codes. They are popular these days because they can be quantum-secure. In this paper, we study the Niederreiter cryptosystem using quasi-cyclic codes. We prove, if these quasi-cyclic codes satisfy certain conditions, the corresponding Niederreiter cryptosystem is resistant to the hidden subgroup problem using quantum Fourier sampling. Our proof requires the classification of finite simple groups