Orbit codes from forms on vector spaces over a finite field
The authors establish new techniques for the construction of orbit codes in this paper. They construct different families of orbit codes in the vector spaces of the symmetric bilinear forms, quadratic forms and Hermitian forms on an \(n\)-dimensional vector space over the finite field \(\mathbb{F}_q\). And also these codes admit the general linear group as a transitive automorphism group. They point out that \(\mathrm{GL}(V)\) is a subgroup of \(\mathrm{GL}(\tilde V)\). In particular, \(\mathrm{GL}(V)\) acts on the forms by preserving the rank. The authors prove the existing result for \(n\geq 3\), as follows; ``The stabilizer of \(U_2\) in \(\mathrm{GL}(V)\) is \(S\times C\), where \(S\) is the Singer cyclic group of \(\mathrm{GL}(V)\) generated by \(\sigma\) and \(C\) is the cyclic group generated by \(\tau\)''. Particularly, the authors refer to a good number of material and present them as references perfectly.
scientific article; zbMATH DE number 7461991
scientific article