Kneser-Hecke-operators in coding theory

The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. It maps a linear self-dual code $C$ over a finite field to the formal sum of the equivalence classes of those self-dual codes that intersect $C$ in a codimension 1 subspace. The eigenspaces of this self-adjoint linear operator may be described in terms of a coding-theory analogue of the Siegel $\Phi$-operator

On extremal self-dual ternary codes of length 48

All extremal ternary codes of length 48 that have some automorphism of prime order $p\geq 5$ are equivalent to one of the two known codes, the Pless code or the extended quadratic residue code

On self-dual MRD codes

We determine the automorphism group of Gabidulin codes of full length and characterise when these codes are equivalent to self-dual codes.Comment: Improved exposition according to the referees' comment

Hermitian modular forms congruent to 1 modulo p

For any natural number $\ell$ and any prime $p\equiv 1 \pmod{4}$ not dividing $\ell$ there is a Hermitian modular form of arbitrary genus $n$ over L:=\Q [\sqrt{-\ell}] that is congruent to 1 modulo $p$ which is a Hermitian theta series of an $O_L$-lattice of rank $p-1$ admitting a fixed point free automorphism of order $p$. It is shown that also for non-free lattices such theta series are modular forms