1,740 research outputs found
Probabilistic lower bounds on maximal determinants of binary matrices
Let be the maximal determinant for -matrices, and be the ratio of
to the Hadamard upper bound. Using the probabilistic method,
we prove new lower bounds on and in terms of
, where is the order of a Hadamard matrix and is maximal
subject to . For example, if , and if . By a recent result of Livinskyi, as ,
so the second bound is close to for large . Previous
lower bounds tended to zero as with fixed, except in the
cases . For , our bounds are better for all
sufficiently large . If the Hadamard conjecture is true, then , so
the first bound above shows that is bounded below by a positive
constant .Comment: 17 pages, 2 tables, 24 references. Shorter version of
arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated
references in v5, added Remark 2.8 and a reference in v6, updated references
in v
Embedding cocylic D-optimal designs in cocylic Hadamard matrices
A method for embedding cocyclic submatrices with âlargeâ determinants of orders
2t in certain cocyclic Hadamard matrices of orders 4t is described (t an odd integer). If these
determinants attain the largest possible value, we are embedding D-optimal designs. Applications
to the pivot values that appear when Gaussian elimination with complete pivoting is performed on
these cocyclic Hadamard matrices are studied.Ministerio de Ciencia e InnovaciĂłn MTM2008-06578Junta de AndalucĂa FQM-016Junta de AndalucĂa P07-FQM-0298
Determinants of (â1,1)-matrices of the skew-symmetric type: a cocyclic approach
An n by n skew-symmetric type (â1, 1)-matrix K = [ki,j ] has 1âs on the main
diagonal and ±1âs elsewhere with ki,j = âkj,i. The largest possible determinant of such
a matrix K is an interesting problem. The literature is extensive for n 0 mod 4 (skew-
Hadamard matrices), but for n 2 mod 4 there are few results known for this question.
In this paper we approach this problem constructing cocyclic matrices over the dihedral
group of 2t elements, for t odd, which are equivalent to (â1, 1)-matrices of skew type.
Some explicit calculations have been done up to t = 11. To our knowledge, the upper
bounds on the maximal determinant in orders 18 and 22 have been improved.Junta de AndalucĂa FQM-01
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