Bounds for flag codes

The target value of a feasible solution of an optimization problem is the value of the function that is optimized evaluated at that point. In the ILP of Proposition 3.12 the target function is the sum on the right hand side of (6).

By an optimal target value we denote the target value that is attained in the extremum, i.e., the maximum or minimum depending on the formulation of the optimization problem.

Let us assume \(A_2^f(6,3;\{2,3\})\le 185\) for a moment. Inequality (10) then would yield \(A_2^f(7,3;\{3,4\})\le \frac{127}{7}\cdot 185=3356 +\frac{3}{7}\). Of course this can be rounded down to 3356, since \(A_2^f(7,3;\{3,4\})\) is an integer. However, as in the case of constant dimension codes the rounding of the Johnson bound can be improved using the theory of \(q^r\)-divisible codes, see [10]. More concretely, for each codeword \(\left( W_3,W_4\right) \) we just consider the plane \(W_3\). Since we assume \(A_2^f(6,3;\{2,3\})\le 185\) those planes cover each point of \({\mathbb {F}}_2^7\) at most 185 times. If the flag code has cardinality 3356 then not every point of \({\mathbb {F}}_2^7\) can be covered exactly 185 times, i.e., the missing points correspond to a multiset of points of cardinality 3, which in turn corresponds to a binary linear code of effective length 3. Since it can be shown that this code has to be 4-divisible, i.e., the weight of every codeword has to be divisible by 4 and such a code cannot exist, we could strengthen our argument to \(A_2^f(7,3;\{3,4\})\le 3355\). (A 4-divisible binary linear code of effective length 10 indeed exists.) For the details we refer to [10, Lemma 13(i)] and its preparing results and definitions.