1. **Problem statement:** We have 9 binary vectors of length 8, and we consider their 36 mutual sums modulo 2 (i.e., sums of pairs of distinct vectors, component-wise mod 2). We want to know if these 36 sums can all be distinct and form an antichain (no vector is component-wise less than or equal to another). 2. **Key concepts:** - The sum mod 2 of two binary vectors is their bitwise XOR. - An antichain in the Boolean lattice $\{0,1\}^8$ is a set of vectors where no vector is less than or equal to another in all coordinates. - The number of pairs from 9 vectors is $\binom{9}{2} = 36$. 3. **Important facts:** - The maximum size of an antichain in $\{0,1\}^8$ is given by the largest binomial coefficient $\binom{8}{4} = 70$. - So 36 vectors can form an antichain in principle. 4. **Constraints on sums:** - The sums are pairwise XORs of the original 9 vectors. - All 36 sums must be distinct. - The sums must form an antichain. 5. **Analysis:** - Since the sums are XORs of pairs, the sums are symmetric differences of the supports of the vectors. - The original 9 vectors must be chosen so that all pairwise XORs are distinct. - Also, these sums must be incomparable under the partial order defined by component-wise $\leq$. 6. **Feasibility:** - Distinctness of sums is possible if the original vectors are chosen carefully. - However, forming an antichain from all sums is very restrictive because XOR tends to produce vectors with varying weights and supports. - The antichain condition means no sum vector is contained in another. 7. **Conclusion:** - It is theoretically possible to have 36 distinct sums forming an antichain in $\{0,1\}^8$ because the maximum antichain size is 70. - But whether 9 vectors can be chosen so that all 36 sums are distinct and form an antichain is a nontrivial combinatorial design problem. - Without explicit construction or known results, the problem remains open. **Final answer:** It is possible in principle, but no known explicit construction guarantees 9 vectors of length 8 whose 36 pairwise sums mod 2 are all distinct and form an antichain.