<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><span class="Apple-tab-span" style="white-space:pre"> </span>Department of Computer Science/The University of Chicago<div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span>*** Dissertation Defense ***</div><div class=""><br class=""></div><div class=""><span class="Apple-tab-span" style="white-space:pre"> </span>Joint PhD program in Mathematics and Computer Science</div><div class=""><br class=""></div><div class=""><br class=""></div><div class="">Candidate: Angela Wu</div><div class=""><br class=""></div><div class="">Date: Friday, April 27, 2018</div><div class=""><br class=""></div><div class="">Time: 2:30 pm</div><div class=""><br class=""></div><div class="">Place: Ryerson 277</div><div class=""><br class=""></div><div class="">Title: List Decoding Homomorphism Codes</div><div class=""><br class=""></div><div class="">Abstract:</div><div class="">
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<!--StartFragment--><p class="MsoNormal"><span lang="EN" class="">The codewords of the <i class="">homomorphism code</i> aHom(G,H)
are the affine homomorphisms between two finite groups, G and H,
generalizing Hadamard codes. Following
the work of Goldreich--Levin (1989), Grigorescu et al. (2006), Dinur et al.
(2008), and Guo and Sudan (2014), who
demonstrated local list-decodability up to minimum distance of homomorphism
codes for expanding classes of groups (Boolean, abelian, nilpotent). We further
expand the range of groups with this property. In particular, for the first
time, we do not require either G or H to be solvable. Specifically, we
demonstrate a poly(1/𝜀) bound on the list size, i.e., the number of
codewords within distance (mindist-𝜀) from any received word, when G is
an alternating group, and <b class="">H is an
arbitrary (finite or infinite) group</b>. We conjecture that a similar bound
holds for all finite simple groups as domains; the alternating groups serve as
the first test case.</span></p><p class="MsoNormal"><span lang="EN" class="">Our main result is efficient local
list-decoding for the permutation representations of alternating groups (i.e.,
when the codomain is a symmetric group S<sub class="">m</sub>) under the restriction
that the domain G=A<sub class="">n</sub> is paired with codomain H=S<sub class="">m</sub></span><span lang="EN" style="font-family:"Arial Unicode MS",sans-serif" class=""> satisfying m <
2^(n-1)/√n.</span> </p><p class="MsoNormal"><span lang="EN" class="">The limitations on the codomain in the latter
case reflect a gap between uniquely identifying a homomorphism in aHom(A<sub class="">n</sub>,H)
and determining the homomorphism on generators of the whole group. This
phenomenon is new and is sure to appear again for other more general classes of
domains. Bridging this gap requires solving the <b class="">Homomorphism Extension Problem</b> (HomExt): given a partial map 𝛾:
G ⇀ H (the domain of 𝛾 is a subset of G) decide whether or not there
exists a homomorphism 𝜑: G → H
extending 𝛾.</span></p><p class="MsoNormal"><span lang="EN" class="">For this reason, we introduce an intermediate
algorithmic model we call <b class="">Certificate
List-Decoding</b> that avoids the HomExt bottleneck and works in the
alternating versus arbitrary setting.</span></p><p class="MsoNormal"><span lang="EN" style="font-family:"Arial Unicode MS",sans-serif" class="">Our
new combinatorial tools allow us to play on the relatively well-understood top
layers of the subgroup lattice of the domain, avoiding the dependence on the
codomain, a bottleneck in previous work. We also introduce
``mean-list-decoding,'' a relaxation principle for constraints on the domain,
that automatically upgrades results such as {abelian→abelian} to
{arbitrary→abelian}.</span></p><p class="MsoNormal"><span lang="EN" class="">While motivated by bridging the mentioned gap
in list-decoding, HomExt is also of independent interest, both as a problem in
computational group theory and as a new and natural problem in NP of unsettled
complexity status.<o:p class=""></o:p></span></p><p class="MsoNormal">We consider the case H=S<sub class="">m</sub> (the
symmetric group of degree m), i.e., 𝛾 : G ⇀ H gives a group action by
the subgroup generated by the domain of 𝛾. We assume G ≤ S<sub class="">n</sub>
is given as a permutation group by a list of generators. We characterize the
equivalence classes of extensions in terms of a multidimensional oracle
subset-sum problem. From this we infer
that for bounded G the HomExt problem can be solved in polynomial time.</p><p class="MsoNormal"><span lang="EN" class="">We are most concerned with the case G=A<sub class="">n</sub>
(the alternating group of degree n) for variable n under the assumption that
the index of M in G is bounded by poly(n).
We solve this case in polynomial time for all m < 2<sup class="">n-1</sup></span><span lang="EN" style="font-family: 'Arial Unicode MS', sans-serif;" class="">/√n. This is the case required for the main
list-decoding result.</span></p><p class="MsoNormal">The Homomorphism Extension results are solo
work; the rest are joint with Laszlo Babai and Timothy Black.</p><p class="MsoNormal"><span lang="EN" class=""> </span></p><div class="">Angela’s advisor is Prof. Laszlo Babai</div><div class=""><br class=""></div><div class="">A copy of Angela Wu’s dissertation draft is available at this link: <a href="http://math.uchicago.edu/~wu/thesis/thesis.pdf" class="">http://math.uchicago.edu/~wu/thesis/thesis.pdf</a></div><div class=""><br class=""></div><div class=""><br class=""></div></div><div class=""><div class="">
<div class="">=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=</div><div class="">Margaret Jaffey</div><div class="">Department of Computer Science</div><div class="">Student Affairs Administrator</div><div class=""><a href="mailto:margaret@cs.uchicago.edu" class="">margaret@cs.uchicago.edu</a></div><div class="">Eckhart 124</div><div class="">773-702-6011</div><div class="">=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=</div><div class=""><br class=""></div><br class="Apple-interchange-newline">
</div>
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