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A planar Eulerian triangulation is a simple plane graph in which each face is a triangle and each vertex has even degree. Such objects are known to be equivalent to spherical Latin bitrades. (A Latin bitrade describes the difference between two Latin squares of the same order.) We give a classification in the near-regular case when each vertex is of degree or (which we call a near-homogeneous spherical Latin bitrade, or NHSLB). The classification demonstrates that any NHSLB is equal to two graphs...
* This work was partially supported by the Bulgarian National Science Fund under Contract No. MM – 503/1995.New extremal binary self-dual codes of lengths 50 and 52 are
constructed. Some of them are the first known codes with such weight
enumerators. The structure of their automorphisms groups are shown.
∗ This work has been partially supported by the Bulgarian NSF under Contract No. I-506/1995.In this note we construct five new symmetric 2-(61,16,4) designs
invariant under the dihedral group of order 10. As a by-product we
obtain 25 new residual 2-(45,12,4) designs. The automorphism groups of all
new designs are computed.
The maximal cardinality of a code W on the unit sphere in n dimensions
with (x, y) ≤ s whenever x, y ∈ W, x 6= y, is denoted by A(n, s). We use two
methods for obtaining new upper bounds on A(n, s) for some values of n and s.
We find new linear programming bounds by suitable polynomials of degrees which
are higher than the degrees of the previously known good polynomials due to
Levenshtein [11, 12]. Also we investigate the possibilities for attaining the Levenshtein
bounds [11, 12]. In such cases...
A loop of order possesses at least associative triples. However, no loop of order that achieves this bound seems to be known. If the loop is involutory, then it possesses at least associative triples. Involutory loops with associative triples can be obtained by prolongation of certain maximally nonassociative quasigroups whenever is a prime greater than or equal to or , an odd prime. For orders the minimum number of associative triples is reported for both general and involutory...
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