OKlibrary  0.2.1.6
SelfDuality.hpp File Reference

On investigations regarding self-dual {0,1}-matrices. More...

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Detailed Description

On investigations regarding self-dual {0,1}-matrices.

Todo:
Smallest self-dual but not self-polar matrix
  • See the "Experiment" in ComputerAlgebra/CombinatorialMatrices/Lisp/Isomorphisms.mac.
  • Interesting also to determine the number of self-dual and self-polar matrices.
  • Easy cases (showing the quotient #self-dual/#all):
    1. n=0: 1/1
    2. n=1: 2/2
    3. n=2: 12/16
    4. n=3: 248/512
    5. n=4: 15428/65536
    In each case every self-dual matrix was self-polar.
  • n = 5:
    1. Random sampling yields 16686/213822.
    2. By a C++ implementation we could run through all cases (see OKlib/Combinatorics/plans/general.hpp).
  • n = 6:
    1. Random sampling: 944/58092.
  • n = 7:
    1. Random sampling: 15/6060.
    2. Another run with set_random(0): 171/100037.
  • n = 8:
    1. With default setting: 0/2567.
    2. With set_random(0): 1/2976 (segfault at interrupt).
    3. With set_random(1): 2/11720 (with Ecl; after several interrupts lost connection).
  • It seems that for small n (n <= 5) there are no self-dual but not self-polar matrices, while for bigger n self-dual matrices become very rare (so that a random search for them won't succeed).

Definition in file SelfDuality.hpp.