New upper bounds for the football pool problem for 11 and 12 matches

AbstractWe consider the problem of minimizing the number of words in a code with the property that all words in the space F3n are within Hamming distance 1 from some codeword. This problem is called the football pool problem, since the words in such a code can be used in a football pool to guarantee that at least one forecast has at least n − 1 correct results. In this note we show that for 11 and 12 matches, there are 9477 and 27702 words, respectively, having the aforementioned property. Simulated annealing has played an important role in the search for these words

The bacterial antitoxin HipB establishes a ternary complex with operator DNA and phosphorylated toxin HipA to regulate bacterial persistence

Nearly all bacteria exhibit a type of phenotypic growth described as persistence that is thought to underlie antibiotic tolerance and recalcitrant chronic infections. The chromosomally encoded high-persistence (Hip) toxin-antitoxin proteins HipA(SO) and HipB(SO) from Shewanella oneidensis, a proteobacterium with unusual respiratory capacities, constitute a type II toxin-antitoxin protein module. Here we show that phosphorylated HipA(SO) can engage in an unexpected ternary complex with HipB(SO) and double-stranded operator DNA that is distinct from the prototypical counterpart complex from Escherichia coli. The structure of HipB(SO) in complex with operator DNA reveals a flexible C-terminus that is sequestered by HipA(SO) in the ternary complex, indicative of its role in binding HipA(SO) to abolish its function in persistence. The structure of HipA(SO) in complex with a non-hydrolyzable ATP analogue shows that HipA(SO) autophosphorylation is coupled to an unusual conformational change of its phosphorylation loop. However, HipA(SO) is unable to phosphorylate the translation factor Elongation factor Tu, contrary to previous reports, but in agreement with more recent findings. Our studies suggest that the phosphorylation state of HipA is an important factor in persistence and that the structural and mechanistic diversity of HipAB modules as regulatory factors in bacterial persistence is broader than previously thought

Bounds on codes over an alphabet of five elements

AbstractWe consider the problem of finding bounds and exact values of A5(n,d) — the maximum size of a code of length n and minimum distance d over an alphabet of 5 elements. Using a wide variety of constructions and methods, a table of bounds on A5(n,d) for n⩽11 is obtained

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems

Switching codes and designs

AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters

Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page