On metric properties of maps between Hamming spaces and related graph homomorphisms

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• 作者：Yury Polyanskiy ^{yp@mit.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Error-correcting codes ; Graph homomorphism ; Schrijver's θ-function ; Projective geometry over F2F2
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：227-251
• 全文大小：532 K

文摘

A mapping of k-bit strings into n  -bit strings is called an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si1.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=192a5ab18adce2b778d6f07807cbf876" title="Click to view the MathML source">(α,β)class="mathContainer hidden">class="mathCode">$(\alpha ,\beta )$-map if k-bit strings which are more than αk apart are mapped to n-bit strings that are more than βn   apart in Hamming distance. This is a relaxation of the classical problem of constructing error-correcting codes, which corresponds to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si2.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=671e89c670579c8548e73fdd0b138e22" title="Click to view the MathML source">α=0class="mathContainer hidden">class="mathCode">$\alpha =0$. Existence of an class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si1.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=192a5ab18adce2b778d6f07807cbf876" title="Click to view the MathML source">(α,β)class="mathContainer hidden">class="mathCode">$(\alpha ,\beta )$-map is equivalent to existence of a graph homomorphism class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si3.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=d046a8890d3e60ac4d9311ff9ce82295">class="imgLazyJSB inlineImage" height="19" width="152" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si3.gif">class="mathContainer hidden">class="mathCode">$\overline{H}(k,\alpha k)\to \overline{H}(n,\beta n)$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si4.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=a60d896e277f97b7752b5bef4cc5406e" title="Click to view the MathML source">H(n,d)class="mathContainer hidden">class="mathCode">$H(n,d)$ is a Hamming graph with vertex set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si5.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=6f11b9d65e23bfcbbbbe454d496e3001" title="Click to view the MathML source">{0,1}ⁿclass="mathContainer hidden">class="mathCode">${\{0,1\}}^n$ and edges connecting vertices differing in d or fewer entries.

This paper proves impossibility results on achievable parameters class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si1.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=192a5ab18adce2b778d6f07807cbf876" title="Click to view the MathML source">(α,β)class="mathContainer hidden">class="mathCode">$(\alpha ,\beta )$ in the regime of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si6.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=171462487b5900edf6f3639ca9de83aa" title="Click to view the MathML source">n,k→∞class="mathContainer hidden">class="mathCode">$n,k\to \infty$ with a fixed ratio class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si352.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=b802b2eebcab5acb8d66b8463482d050">class="imgLazyJSB inlineImage" height="17" width="43" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si352.gif">class="mathContainer hidden">class="mathCode">$\frac{n}{k}=\rho$. 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si112.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=522846aaf5350e1e92ca3f73ff7f3605" title="Click to view the MathML source">β>1/2class="mathContainer hidden">class="mathCode">$\beta >1/2$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si9.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=11c7f5963e5721e6cb3320db4d3f02af">class="imgLazyJSB inlineImage" height="16" width="11" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si9.gif">class="mathContainer hidden">class="mathCode">$\frac{n}{k}$ – integer, the class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si9.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=11c7f5963e5721e6cb3320db4d3f02af">class="imgLazyJSB inlineImage" height="16" width="11" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516300772-si9.gif">class="mathContainer hidden">class="mathCode">$\frac{n}{k}$-fold repetition map achieving class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si10.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=009b3b12fb5e784d65b46d6ad4df2ffb" title="Click to view the MathML source">α=βclass="mathContainer hidden">class="mathCode">$\alpha =\beta$ is asymptotically optimal.

Finally, constraints on configurations of points and hyperplanes in projective spaces over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0097316516300772&_mathId=si11.gif&_user=111111111&_pii=S0097316516300772&_rdoc=1&_issn=00973165&md5=69a1a114ff17eca1376c34506c550557" title="Click to view the MathML source">F₂class="mathContainer hidden">class="mathCode">${\mathbb{F}}_2$ are derived.