Inspection of optimum Golomb squares found by exhaustive computer search shows some are symmetric with respect to diagonal reflection. This suggests searching for such symmetric Golomb squares might be an efficient way to find good squares. In this correspondence, we find the best symmetric N/spl times/N Golomb squares for N/spl les/22. Some appear to be the best known.

Let a Golomb ruler be a set {a/sub i/} of integers so that all the differences a/sub i/-a/sub j/, i/spl ne/j, are distinct. Let H(I.J) be the smallest n such that there are I disjoint Golomb rulers each containing J elements chosen from {1,2,...n}. In 1990, Klove gave a table of bounds on H(I,J). In this correspondence we improve and extend this table with results found by computer search.

For an undirected graph and an optimal cyclic list of all its vertices, the cyclic cover time is the expected time it takes a simple random walk to travel from vertex to vertex along the list until it completes a full cycle. The main result of this paper is a characterization of the cyclic cover time in terms of simple and easy-to-compute graph properties. Namely, for any connected graph, the cyclic cover time is $\Theta (n^2 d_{ave} (d^{ - 1} )_{ave} $), where n is the number of vertices in the graph, $d_{ave} $ is the average degree of its vertices, and $(d^{ - 1} )_{ave} $ is the average of the inverse of the degree of its vertices. Other results obtained in the processes of proving the main theorem are a similar characterization of minimum resistance spanning trees of graphs, improved bounds on the cover time of graphs, and a simplified proof that the maximum commute time in any connected graph is at most $4n^3 /27 + o(n^3 )$.

A table of binary constant weight codes of length n>

By exhaustive computer search, the minimum length Golomb rulers (or B/sub 2/-sequences or difference triangles) containing 14, 15, and 16 marks are found. They are unique and of length 127, 151, and 177, respectively.>

A high-resolution raster-graphics display is usually combined with processing power and a memory organization that facilitates basic graphics operations. For many applications, including interactive text processing, the ability to quickly move or copy small rectangles of pixels is essential. This paper proposes a novel organization of raster-graphics memory that permits all small rectangles to be moved efficiently. The memory organization is based on a doubly periodic assignment of pixels to M memory chips according to a “Fibonacci” lattice. The memory organization guarantees that, if a rectilinearly oriented rectangle contains fewer than M / @@@@5 pixels, then all pixels will reside in different memory chips and thus can be accessed simultaneously. Moreover, any M consecutive pixels, arranged either horizontally or vertically, can be accessed simultaneously. We also define a continuous analog of the problem, which can be posed as: “What is the maximum density of a set of points in the plane such that no two points are contained in the interior of a rectilinearly oriented rectangle of unit area?” We show the existence of such a set with density 1/ @@@@5, and prove this is optimal by giving a matching upper bound.

An example is provided of two convolutional codes that have the same transmission gain but whose dual codes do not. This shows that no analog of the MacWilliams identity for block codes can exist relating the transmission gains of a convolutional code and its dual.