Given two$q$-ary codes$C_{1}$and$C_{2}$, the relative hull of$C_{1}$with respect to$C_{2}$is the intersection$C_{1}\cap C_{2}^{\perp} $. We prove that when$q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the$e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when$q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.

The orientable domination number, DOM(G), of a graph G is the largest domination number over all orientations of G. In this paper, DOM is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. (1996) is extended by establishing the values of DOM(Kn1,n2,n3) for arbitrary positive integers n1,n2 and n3. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in Brešar et al. (2022).

We consider storage systems in which K files are stored over N nodes. A node may be systematic for a particular file in the sense that access to it gives access to the file. Alternatively, a node may be coded, meaning that it gives access to a particular file only when combined with other nodes (which may be coded or systematic). Requests for file fkarrive at rate λk, and we are interested in the rate that can be served by a particular system. In this paper, we determine the set of request arrival rates for the a 3-file coded storage system. We also provide an algorithm to maximize the rate of requests served for file K given λ1, . . . , λK-1in a general K-file case.

Combinatorial structures called stopping sets are useful in analyzing the performance of a linear code when coupled with an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of Hermitian codes.

To the overarching question in this special issue as to whether it is wise to pursue an inclusive pluralist approach to dynamical and symbolic theoretical frameworks in cognitive science, Rick Dale...

According to today's best projections, petaFLOP/s computing platforms will combine deep memory hierarchies in both latency and bandwidth with a need for many-thousand-fold parallelism. Unless effective parallel programs are prepared in advance, much of the promise of the first year or two of operation for these systems may be lost. We introduce a candidate for a portable petaFLOP/s programming model that can enable these important early application programs to be developed while, at the same time, permitting these same applications to run efficiently on the most capable computing systems now available. An MPI-based model is portable, but its programming paradigm ignores the potential benefits of hardware support for shared memory within each network node. A threads-based model cannot directly cope with the distributed nature of the memory over the network. Therefore, a new, portable programming model is needed. The shared memory programming model dramatically simplifies the expression of dynamic load balancing strategies for irregular algorithms. The main strategy is a transparent self-scheduled task list performed in parallel so long as specified data-dependent conditions are met. The model used is a cluster of multiprocessor distributed shared memory machines with network-attached disks. Our experimental run-time system allows the programmer to view this computing platform as a single machine with a four-stage memory hierarchy, consisting of coherent processor cache, non-coherent local shared memory, global shared memory, plus a global disk file system.

Understanding turbulence and mix in compressible flows is of fundamental importance to real-world applications such as chemical combustion and supernova evolution.The ability to run in three dimensions and at very high resolution is required for the simulation to accurately represent the interaction of the various length scales, and consequently, the reactivity of the intermixing species.Toward this end, we have carried out a very high resolution (over 8 billion zones) 3-D simulation of the Richtmyer-Meshkov instability and turbulent mixing on the IBM Sustained Stewardship TeraOp (SST) system, developed under the auspices of the Department of Energy (DOE) Accelerated Strategic Computing Initiative (ASCI) and located at Lawrence Livermore National Laboratory.We have also undertaken an even higher resolution proofof-principle calculation (over 24 billion zones) on 5832 processors of the IBM system, which executed for over an hour at a sustained rate of 1.05 Tflop/s, as well as a short calculation with a modified algorithm that achieved a sustained rate of 1.18 Tflop/s.The full production scientific simulation, using a further modified algorithm, ran for 27,000 timesteps in slightly over a week of wall time using 3840 processors of the IBM system, clocking a sustained throughput of roughly 0.6 teraflop per second (32-bit arithmetic).Nearly 300,000 graphics files comprising over three terabytes of data were produced and post-processed.The capability of running in 3-D at high resolution enabled us to get a more accurate and detailed picture of the fluidflow structure -in particular, to simulate the development of fine scale structures from the interactions of long-and short-wavelength phenomena, to elucidate differences between twodimensional and three-dimensional turbulence, to explore a conjecture regarding the transition from unstable flow to fully developed turbulence with increasing Reynolds number, and to ascertain convergence of the computed solution with respect to mesh resolution.