arXiv subjects
Publications and source records attributed to Bernardo G. Rodrigues.
We develop methods for determining key properties (simplicity and the dimension of radical) of flip subalgebras in Matsuo algebras. These are interesting classes of commutative non-associative algebras that were introduced within the broader paradigm of axial algebras.
Let $ x $ be an element of a finite group $ G $ and denote the order of $ x $ by $ \mathrm{ord}(x) $. We consider a finite group $ G $ such that $ \gcd(\mathrm{ord}(x),\mathrm{ord}(y))\leqslant 2 $ for any two vanishing elements $ x $ and $ y $ contained in distinct conjugacy classes. We show that such a group $ G $ is solvable. When $ G $ with the property above is supersolvable, we show that $ G $ has a normal metabelian $ 2 $-complement.
A binary $[98280, 24, 47104]_2$ projective two-weight code related to the sporadic simple group ${\rm Co}_1$ of Conway is constructed as a faithful and absolutely irreducible submodule of the permutation module induced by the primitive action of ${\rm Co}_1$ on the cosets of ${\rm Co}_2$. The dual code of this code is a uniformly packed $[98280, 98256,3]_2$ code. The geometric significance of the codewords of the code can be traced to the vectors in the Leech lattice, thus revealing that the stabilizer of any non-zero weight codeword in the code is a maximal subgroup of ${\rm Co}_1$. Similarly, the stabilizer of the codewords of minimum weight in the dual code is a maximal subgroup of ${\rm Co}_1$. As by-product, a new strongly regular graph on 16777216 vertices and valency 98280 is constructed using the codewords of the code.