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The problem of finding packings of congruent circles in a circle, or, equivalently, of spreading points in a circle, is considered. Two packing algorithms are discussed, and the best packings found of up to 65 circles are presented.
Let A3(n, d,w) denote the maximum cardinality of a ternary code with length n, minimum distance d, and constant Hamming weight w. Methods for proving upper and lower bounds on A3(n, d,w) are presented, and a table of exact values and bounds in the range n ≤ 10 is given.
The Hungarian mathematician Farkas Bolyai (1775–1856) published in his principal work (‘Tentamen’, 1832–33 [Bol04]) a dense regular packing of equal circles in an equilateral triangle (see Fig. 1). He defined an infinite packing series and investigated the limit of vacuitas (in Latin, the gap in the triangle outside the circles). It is interesting that… (More)
Upper and lower bounds are presented for the maximal possible size of mixed binary/ternary error-correcting codes. A table up to length 13 is included. The upper bounds are obtained by applying the linear programming bound to the product of two association schemes. The lower bounds arise from a number of different constructions.
A binary code <i>C</i> ⊆ F<sub>2</sub><i>n</i> with minimum distance at least <i>d</i> and codewords of Hamming weight <i>w</i> is called an <i>(n</i>,<i>d</i>,<i>w</i>) constant weight code. The maximum size of an <i>(n</i>,<i>d</i>,<i>w</i>) constant weight code is denoted by <i>A</i>(<i>n</i>,<i>d</i>,<i>w</i>), and codes of this size are said to… (More)
The maximum number of codewords in a binary code with length n and minimum distance d is denoted by A(n; d). By construction it is known that A(10; 3) 72 and A(11; 3) 144. These bounds have long been conjectured to be the exact values. This is here proved by classifying various codes of smaller length and lengthening these using backtracking and isomorphism… (More)