Bounds for flag codes (Q2243893)

The Grassmann distance of two flags \((W_1,\ldots , W_m)\) and \((V_1,\ldots , V_m)\) of the same Type \(T=(\dim (W_i))_{i=1..m} = (\dim(V_i))_{i=1..m} \) is defined as \(\sum _{i=1}^m (\dim(W_i) - \dim(W_i \cap V_i) )\), i.e. the sum of the Grassmann distances of the subspaces in the flags. The paper derives upper and lower bounds on the number \(A_q^f(v,d,T)\) of flags in \({\mathbb F}_q^v\) of a given type \(T\) of minimum distance \(\geq d\). Most of the bounds are for full flags, where \(A_q^f(v,d) := A_q^f(v,d,(1,\ldots, v) )\). For certain parameters, the author obtains exact values: \begin{itemize} \item \(A_q^f(3,2) = q^2+q+1 \) \item \(A_q^f(4,3) = q^3+q^2+q+1 \) \item \(A_q^f(v,1) = \prod_{i=2}^v \frac{q^i-1}{q-1} \) \item \(A_q^f(2k,k^2) = q^k+1 \) \item \(A_q^f(2k+1,k^2+k) = q^{k+1}+1 \) \end{itemize} Note that these values are messed up in the statements of the propositions in the printed paper. The methods of proofs are versatile; the author uses graph theoretic methods to obtain good integer linear programming bounds as well as explicit computer constructions (partly with a prescribed symmetry group) to obtain good lower bounds.