This paper proves impossibility results on achievable parameters 0772&_mathId=si1.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=192a5ab18adce2b778d6f07807cbf876" title="Click to view the MathML source">(α,β) in the regime of 0772&_mathId=si6.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=171462487b5900edf6f3639ca9de83aa" title="Click to view the MathML source">n,k→∞ with a fixed ratio 0772&_mathId=si352.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=b802b2eebcab5acb8d66b8463482d050">0772-si352.gif">. This is done by developing a general criterion for existence of graph-homomorphism based on the semi-definite relaxation of the independence number of a graph (known as the Schrijver's θ-function). The criterion is then evaluated using some known and some new results from coding theory concerning the θ -function of Hamming graphs. As an example, it is shown that if 0772&_mathId=si112.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=522846aaf5350e1e92ca3f73ff7f3605" title="Click to view the MathML source">β>1/2 and 0772&_mathId=si9.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=11c7f5963e5721e6cb3320db4d3f02af">
0772-si9.gif"> – integer, the 0772&_mathId=si9.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=11c7f5963e5721e6cb3320db4d3f02af">
0772-si9.gif">-fold repetition map achieving 0772&_mathId=si10.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=009b3b12fb5e784d65b46d6ad4df2ffb" title="Click to view the MathML source">α=β is asymptotically optimal.
Finally, constraints on configurations of points and hyperplanes in projective spaces over 0772&_mathId=si11.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=69a1a114ff17eca1376c34506c550557" title="Click to view the MathML source">F2 are derived.