Conjugate weight enumerators and invariant theory

The Galois group of a finite field extension $K/F$ defines a grading on the symmetric algebra of the $F$-space $K^v$ which we use to introduce the notion of homogeneous conjugate invariants for subgroups $G\leq \GL_v(K)$. If the Weight Enumerator Conjecture holds for a finite representation $\rho $ then the genus-$m$ conjugate complete weight enumerators of self-dual codes generate the corresponding space of conjugate invariants of the associated genus-$m$ Clifford-Weil group ${\mathcal C}_m(\rho ) \leq \GL_{v^m}(K)$. This generalisation of a paper by Bannai, Oura and Da Zhao provides new examples of Clifford-Weil orbits that form projective designs