557 research outputs found
Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms
We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric designs. We prove that -geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric designs
Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms
We show that there are exactly 4285 symmetric (45,12,3) designs that admit nontrivial automorphisms. Among them there are 1161 self-dual designs and 1562 pairs of mutually dual designs. We describe the full automorphism groups of these designs and analyze their ternary codes. R. Mathon and E. Spence have constructed 1136 symmetric (45,12,3) designs with trivial automorphism group, which means that there are at least 5421 symmetric (45,12,3) designs. Further, we discuss trigeodetic graphs obtained from the symmetric designs. We prove that -geodetic graphs constructed from mutually non-isomorphic designs are mutually non-isomorphic, hence there are at least 5421 mutually non-isomorphic trigeodetic graphs obtained from symmetric designs
Linear codes with complementary duals from some strongly regular subgraphs of the McLaughlin graph
We describe a number of properties of some ternary linearcodes defined by the adjacency matrices of some stronglyregular graphs that occur as induced subgraphs of the McLaughlin graph, namely the graphs withparameters and respectively. We show that the codes withparameters ,, , , and obtained from these graphs are linear codes with complementary duals and thus meet the asymptotic Gilbert–Varshamov bound
Implementing Brouwer's database of strongly regular graphs
Andries Brouwer maintains a public database of existence results for strongly
regular graphs on vertices. We implemented most of the infinite
families of graphs listed there in the open-source software Sagemath, as well
as provided constructions of the "sporadic" cases, to obtain a graph for each
set of parameters with known examples. Besides providing a convenient way to
verify these existence results from the actual graphs, it also extends the
database to higher values of .Comment: 18 pages, LaTe
Some strongly regular graphs and self-orthogonal codes from the unitary group U4(3)
We construct self-orthogonal codes from the row span over F2 or F3 of the adjacency matrices of some strongly regular graphs defined by the rank-3 action of the simple unitary group U4(3) on the conjugacy classes of some of its maximal subgroups. We establish some properties of these codes and the nature of some classes of codewords
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