5,988 research outputs found
On triply even binary codes
A triply even code is a binary linear code in which the weight of every
codeword is divisible by 8. We show how two doubly even codes of lengths m_1
and m_2 can be combined to make a triply even code of length m_1+m_2, and then
prove that every maximal triply even code of length 48 can be obtained by
combining two doubly even codes of length 24 in a certain way. Using this
result, we show that there are exactly 10 maximal triply even codes of length
48 up to equivalence.Comment: 21 pages + appendix of 10 pages. Minor revisio
Minimal instances for toric code ground states
A decade ago Kitaev's toric code model established the new paradigm of
topological quantum computation. Due to remarkable theoretical and experimental
progress, the quantum simulation of such complex many-body systems is now
within the realms of possibility. Here we consider the question, to which
extent the ground states of small toric code systems differ from LU-equivalent
graph states. We argue that simplistic (though experimentally attractive)
setups obliterate the differences between the toric code and equivalent graph
states; hence we search for the smallest setups on the square- and triangular
lattice, such that the quasi-locality of the toric code hamiltonian becomes a
distinctive feature. To this end, a purely geometric procedure to transform a
given toric code setup into an LC-equivalent graph state is derived. In
combination with an algorithmic computation of LC-equivalent graph states, we
find the smallest non-trivial setup on the square lattice to contain 5
plaquettes and 16 qubits; on the triangular lattice the number of plaquettes
and qubits is reduced to 4 and 9, respectively.Comment: 14 pages, 11 figure
A new class of codes for Boolean masking of cryptographic computations
We introduce a new class of rate one-half binary codes: {\bf complementary
information set codes.} A binary linear code of length and dimension
is called a complementary information set code (CIS code for short) if it has
two disjoint information sets. This class of codes contains self-dual codes as
a subclass. It is connected to graph correlation immune Boolean functions of
use in the security of hardware implementations of cryptographic primitives.
Such codes permit to improve the cost of masking cryptographic algorithms
against side channel attacks. In this paper we investigate this new class of
codes: we give optimal or best known CIS codes of length We derive
general constructions based on cyclic codes and on double circulant codes. We
derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all
be classified in small lengths by the building up construction. Some
nonlinear permutations are constructed by using -codes, based on the
notion of dual distance of an unrestricted code.Comment: 19 pages. IEEE Trans. on Information Theory, to appea
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