
TL;DR
This paper introduces the Homomorphism Extension problem, analyzing its complexity in various cases, and provides polynomial-time solutions for specific group structures relevant to coding theory.
Contribution
It characterizes the extension problem for symmetric and alternating groups, offering polynomial-time algorithms under certain conditions, advancing understanding in computational group theory and coding.
Findings
Polynomial-time solution for bounded G cases
Extension characterization via subset-sum problem
Efficient algorithms for G=A_n with bounded index
Abstract
We define the Homomorphism Extension (HomExt) problem: given a group , a subgroup and a homomorphism , decide whether or not there exists a homomorphism extending , i.e., . This problem arose in the context of list-decoding homomorphism codes but 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. We consider the case (the symmetric group of degree ), i.e., is a -action on a set of elements. We assume 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 the HomExt problem can be…
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Taxonomy
TopicsCoding theory and cryptography · graph theory and CDMA systems · semigroups and automata theory
