Abstract
We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups $G$ and $H$. We prove some general structural results on how the distance behaves with respect to natural group operations, such as passing to subgroups and quotients, and taking products. Our main result is a general formula for the distance when $G$ is solvable or $H$ is nilpotent, in terms of the normal subgroup structure of $G$ as well as the prime divisors of $|G|$ and $|H|$. In particular, we show that in the above case, the distance is independent of the subgroup structure of $H$. We complement this by showing that, in general, the distance depends on the subgroup structure of $H$.
Highlights
1.1 Error correcting codesThe theory of error correcting codes studies codes, which are subsets of Σn for some alphabet Σ and block length n
We investigate the minimum distance of the error correcting code formed by the homomorphisms between two finite groups G and H
Our main result is a general formula for the distance when G is solvable or H is nilpotent, in terms of the normal subgroup structure of G as well as the prime divisors of |G| and |H|
Summary
The theory of error correcting codes studies codes, which are subsets of Σn for some alphabet Σ and block length n. The electronic journal of combinatorics 22(1) (2015), #P1.4 for carefully chosen subcodes the folded Reed-Solomon codes and multiplicity/derivative codes [DL12], the Reed-Muller codes, and abelian group homomorphisms, it was shown that for any constant > 0 one can algorithmically list decode up to ∆ − n errors with a constant list size, depending only on 1/. For all of these codes, the codewords are interpreted as certain functions f : A → B from some domain A to codomain B. This turns out to be a nontrivial problem and serves as the primary motivation of this paper
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