Usually 1-spreads and 1-parallelisms are called line spreads (parallelisms) or just spreads (parallelisms). A t- parallelism is a partition of the set of t-dimensional subspaces by t-spreads. Ī t- spread in the projective space PG( n, q) is a set of distinct t-dimensional subspaces which partition the point set. The basic concepts and notations concerning designs and resolutions, and spreads and parallelisms in projective spaces, can be found, for instance, in. We present the main principles of such a test and point out the cases in which it may not establish that two parallelisms are isomorphic, and therefore the calculation of good invariants might be very helpful.ĭesign Resolutions and Parallelisms of Projective Spaces The invariants are used after the rejection of most of the isomorphic solutions by a normalizer-based minimality test (NM test). We show their effectiveness on some of our recent classifications of parallelisms with a predefined automorphism group. We do not know previous papers presenting them. They can be calculated and compared relatively fast and do not need much memory to be stored. The present paper describes invariants of design resolutions and parallelisms of projective spaces. Very often, however, sensitive invariants need considerable time to be calculated, and a considerable amount of memory to be stored. In a less formal manner, an invariant with relatively high sensitivity is called sensitive. The sensitivity of an invariant is measured by the ratio of the number of classes it distinguishes to the number of non-isomorphic objects under consideration. Structures with different invariants, however, can obviously not be isomorphic to each other, and this is very useful. The invariants that are usually applied are not complete. An invariant is complete if its value is provably different for members of different isomorphism classes. Regardless of the difference in methods, they all make use of suitable invariants, namely functions which yield the same value for all members of an isomorphism class.
#Magmasoft boolean software#
Many authors consider this problem (recently ) and software solving it is available, for instance.
Since the number of isomorphic solutions might be extremely big, classification is usually impossible without an efficient method for their rejection. The approaches to solving the isomorphism problem are of major importance for the success of computer-aided classifications up to isomorphism of various combinatorial structures, such as graphs, designs, design resolutions, codes, Hadamard matrices, etc.
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