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On minimal additive complements of integers
Let . If , then the set is
called an additive complement to in . If no proper subset of
is an additive complement to , then is called a minimal additive
complement. Let . If there exists a positive integer
such that for all sufficiently large integers , then we call
eventually periodic. In this paper, we study the existence of a minimal
complement to when is eventually periodic or not. This partially
answers a problem of Nathanson.Comment: 13 page
A simple family of nonadditive quantum codes
Most known quantum codes are additive, meaning the codespace can be described
as the simultaneous eigenspace of an abelian subgroup of the Pauli group. While
in some scenarios such codes are strictly suboptimal, very little is understood
about how to construct nonadditive codes with good performance. Here we present
a family of nonadditive quantum codes for all odd blocklengths, n, that has a
particularly simple form. Our codes correct single qubit erasures while
encoding a higher dimensional space than is possible with an additive code or,
for n of 11 or greater, any previous codes.Comment: 3 pages, new version with slight clarifications, no results are
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