# Disclosure

Agrell, Vardy & Zeger (*A table of upper bounds for binary codes*,
IEEE Trans. Inf. Th. **47**, 3004-3006, Nov. 2001) write
The bounds A(17,6) ≤ 340, A(21,6) ≤ 4096, A(17,8) ≤ 37,
and A(21,8) ≤ 512 have been derived in [BBMOS], apparently by
linear programming, although the specific inequalities used in
the optimization are not disclosed.

So, let us give this information.
## A(21,8) ≤ 512

Since A(21,8,5) = 21 and A(18,8,5) = 9, we have
A_{16}+12A_{18}+21A_{20} ≤ 21.
(There is at most one word of weight 20, and if there is one
then there are no words of weights 16 or 18. There is at most
one word of weight 18, and if there is one then there are
at most A(18,8,5) = 9 words of weight 16. Otherwise, there are
at most A(21,8,5) = 21 words of weight 16.)
Adding this equation to the standard Delsarte system yields
A(21,8) ≤ 512.
## A(21,6) ≤ 4096

Additional equation: A_{18}+7A_{20} ≤ 7.
## A(17,6) ≤ 340

For an arbitrary code the upper bound found is 341.
For a code of odd size, the upper bound is 340.
Additional equation: A_{14}+5A_{16} ≤ 5.
## A(17,8) ≤ 37

For an arbitrary code the upper bound found is 38.
For a code of size 2 (mod 4), the upper bound is 37.
Additional equation: A_{12}+3A_{14}+7A_{16}≤ 7.
(This follows by A(17,8,5) = 7 and A(14,8,5) = 4.)