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Paramagnetic triplet centers produced by e-beam irradiation of synthetic microcrystalline Ib-type high-pressure high-temperature (HPHT) diamonds were studied by continuous wave (CW) electron paramagnetic resonance (EPR) spectroscopy at X-band (9.4 GHz), pulsed EPR at X- and Q-bands (34 GHz), and fluorescence spectroscopies as a function of radiation fluences up to 5 × 1019e–/cm2. EPR spectra of mostly “forbidden” Δms= 2 electronic spin transitions observed at g≈ 4 (i.e., so-called half-field EPR spectra) reveal the presence of the main W15 triplet defects associated with the fluorescent negatively charged nitrogen-vacancy (NV–) centers as well as additional triplet spin centers identified as W16, W17, W18, and W33 that appear upon increasing the e-beam fluence. Consequent annealing at 1,400 °C significantly reduces the content of W17, W18, and W33 but not W15 and W16 defects. The efficacy of NV–center fabrication as a function of fluence dependent e-beam irradiation is also reported.

The sudoku minimum number of clues problem is the following question: what is the smallest number of clues that a sudoku puzzle can have? For several years it had been conjectured that the answer is 17. We have performed an exhaustive computer search for 16-clue sudoku puzzles and did not find any, thus proving that the answer is indeed 17. In this article we describe our method and the actual search. As a part of this project, we developed a novel way to enumerate hitting sets. The hitting set problem is computationally hard; it is one of Karp’s 21 classic NP-complete problems. A standard backtracking algorithm for finding hitting sets would not be fast enough to search for a 16-clue sudoku puzzle exhaustively, even at today’s supercomputer speeds. To make an exhaustive search possible, we designed an algorithm that allowed us to efficiently enumerate hitting sets of a suitable size.

Abstract: Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66,30,29) and 2 − (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 − (2u2 − u,u2 − u,u2 − u − 1) and 2 − (2u2 + u,u2,u2 − u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey–Rankin bound.

Abstract We characterize the SDP designs that give rise to four-weight spin models with two values. We prove that the only such designs are the symplectic SDP designs. The proof involves analysis of the cardinalities of intersections of four blocks.

We outline the determination of the weight enumerator of the code of the projective plane of order 5 by hand. The main tools used are a version of Gleason's theorem on the enumerators of self-dual codes and geometric descriptions of codewords of low weight. This paper is in part a condensation of the companion paper (Nota di Mat.) that contains full details.

We consider Hadamard matrices with the quasi-3 property, and their implications for the existence of certain quasi-symmetric designs. We find necessary conditions on the parameters, and show that the quasi-3 property is preserved by the Kronecker product.

Certain nonlinear binary codes contain more codewords than any comparable linear code presently known. These include the Kerdock and Preparata codes, which exist for all lengths 4m≥ 16. At length 16 they coincide to give the Nordstrom-Robinson code. This paper constructs a nonlinear (64, 237, 12) code as the binary image, under the Gray map, of an extended cyclic code defined over the integers modulo 4 using Galois rings. The Nordstrom-Robinson code is defined in this same way, and like the Nordstrom-Robinson code, the new code is better than any linear code that is presently known.

Dillon asked whether the all-1 vector is in the binary code of a square (symmetric) design with parameters (22m, 22m−1−2m−1, 22m−2−2m−1) and dimension 2m+2. In this paper we show that the answer to this question is yes. This result gives a characterization of designs with these parameters and minimal 2-rank as SDP designs. Our result also allows us to remove a hypothesis from a theorem of Dillon and Schatz relating difference sets in elementary abelian 2-groups to SDP designs. Along the way we prove results about any designs with the parameters of the residual and derived designs. One of the results deals with a divisibility property that characterizes the elliptic and hyperbolic designs.

We give a characterization of codes meeting the Grey–Rankin bound. When the codes have even length, the existence of such codes is equivalent to the existence of certain quasi-symmetric designs. We also find the parameters of all linear codes meeting the Grey–Rankin bound.

We prove a conjecture of Brouwer, namely that a 2-(28,4,1) design has 2–rank at least 19, with equality occuring if and only if the design is the Ree unital. We give a similar characterization of the Hermitian unital.