Framed vertex operator algebras, codes and the moonshine module

For a simple vertex operator algebra whose Virasoro element is a sum of commutative Virasoro elements of central charge 1/2, two codes are introduced and studied. It is proved that such vertex operator algebras are rational. For lattice vertex operator algebras and related ones, decompositions into direct sums of irreducible modules for the product of the Virasoro algebras of central charge 1/2 are explicitly described. As an application, the decomposition of the moonshine vertex operator algebra is obtained for a distinguished system of 48 Virasoro algebras.Comment: Latex, 54 page

New characterisations of the Nordstrom–Robinson codes

In his doctoral thesis, Snover proved that any binary $(m,256,\delta)$ code is equivalent to the Nordstrom-Robinson code or the punctured Nordstrom-Robinson code for $(m,\delta)=(16,6)$ or $(15,5)$ respectively. We prove that these codes are also characterised as \emph{completely regular} binary codes with $(m,\delta)=(16,6)$ or $(15,5)$, and moreover, that they are \emph{completely transitive}. Also, it is known that completely transitive codes are necessarily completely regular, but whether the converse holds has up to now been an open question. We answer this by proving that certain completely regular codes are not completely transitive, namely, the (Punctured) Preparata codes other than the (Punctured) Nordstrom-Robinson code

The deep space network, volume 18

The objectives, functions, and organization of the Deep Space Network are summarized. The Deep Space Instrumentation Facility, the Ground Communications Facility, and the Network Control System are described

Permutation Decoding and the Stopping Redundancy Hierarchy of Cyclic and Extended Cyclic Codes

We introduce the notion of the stopping redundancy hierarchy of a linear block code as a measure of the trade-off between performance and complexity of iterative decoding for the binary erasure channel. We derive lower and upper bounds for the stopping redundancy hierarchy via Lovasz's Local Lemma and Bonferroni-type inequalities, and specialize them for codes with cyclic parity-check matrices. Based on the observed properties of parity-check matrices with good stopping redundancy characteristics, we develop a novel decoding technique, termed automorphism group decoding, that combines iterative message passing and permutation decoding. We also present bounds on the smallest number of permutations of an automorphism group decoder needed to correct any set of erasures up to a prescribed size. Simulation results demonstrate that for a large number of algebraic codes, the performance of the new decoding method is close to that of maximum likelihood decoding.Comment: 40 pages, 6 figures, 10 tables, submitted to IEEE Transactions on Information Theor

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Perfect binary codes: classification and properties

An r-perfect binary code is a subset of ℤ2n such that for any word, there is a unique codeword at Hamming distance at most r. Such a code is r-error-correcting. Two codes are equivalent if one can be obtained from the other by permuting the coordinates and adding a constant vector. The main result of this thesis is a computer-aided classification, up to equivalence, of the 1-perfect binary codes of length 15. In an extended 1-perfect code, the neighborhood of a codeword corresponds to a Steiner quadruple system. To utilize this connection, we start with a computational classification of Steiner quadruple systems of order 16. This classification is also used to establish the nonexistence of Steiner quintuple systems S(4, 5, 17). The classification of the codes is used for computational examination of their properties. These properties include occurrences of Steiner triple and quadruple systems, automorphisms, ranks, structure of i-components and connections to orthogonal arrays and mixed perfect codes. It is also proved that extended 1-perfect binary codes are equivalent if and only if their minimum distance graphs are isomorphic