List-decoding homomorphism codes with arbitrary codomains

• Main Authors: Babai, László, Black, Timothy J. F, Wuu, Angela
• Format: Journal Article
• Language: English
• Published: 08-06-2018
• Subjects: Computer Science - Computational Complexity; Computer Science - Information Theory; Mathematics - Information Theory
• DOI: 10.48550/arxiv.1806.02969

The codewords of the homomorphism code $\operatorname{aHom}(G,H)$ 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), we further expand
the range of groups for which local list-decoding is possible up to
$\textsf{mindist}$, the minimum distance of the code. In particular, for the
first time, we do not require either $G$ or $H$ to be solvable. Specifically,
we demonstrate a $\operatorname{poly}(1/\varepsilon)$ bound on the list size,
i.e., on the number of codewords within distance
$(\textsf{mindist}-\varepsilon)$ from any received word, when $G$ is either
abelian or an alternating group, and $H$ is an arbitrary (finite or infinite)
group. We conjecture that a similar bound holds for all finite simple groups as
domains; the alternating groups serve as the first test case.
The abelian vs. arbitrary result then permits us to adapt previous techniques
to obtain efficient local list-decoding for this case. We also obtain efficient
local list-decoding for the permutation representations of alternating groups
(i.e., when the codomain is a symmetric group $S_m$) under the restriction that
the domain $G=A_n$ is paired with codomain $H=S_m$ satisfying $m <
2^{n-1}/\sqrt{n}$.
The limitations on the codomain in the latter case arise from severe
technical difficulties stemming from the need to solve the homomorphism
extension (HomExt) problem in certain cases; these are addressed in a separate
paper (Wuu 2018).
However, we also introduce an intermediate "semi-algorithmic" model we call
Certificate List-Decoding that bypasses the HomExt bottleneck and works in the
alternating vs. arbitrary setting. A certificate list-decoder produces partial
homomorphisms that uniquely extend to the homomorphisms in the list.