199,147 research outputs found
Simple approach to approximate quantum error correction based on the transpose channel
We demonstrate that there exists a universal, near-optimal recovery map—the transpose channel—for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement
Coding Theory and Algebraic Combinatorics
This chapter introduces and elaborates on the fruitful interplay of coding
theory and algebraic combinatorics, with most of the focus on the interaction
of codes with combinatorial designs, finite geometries, simple groups, sphere
packings, kissing numbers, lattices, and association schemes. In particular,
special interest is devoted to the relationship between codes and combinatorial
designs. We describe and recapitulate important results in the development of
the state of the art. In addition, we give illustrative examples and
constructions, and highlight recent advances. Finally, we provide a collection
of significant open problems and challenges concerning future research.Comment: 33 pages; handbook chapter, to appear in: "Selected Topics in
Information and Coding Theory", ed. by I. Woungang et al., World Scientific,
Singapore, 201
Characterisation of a family of neighbour transitive codes
We consider codes of length over an alphabet of size as subsets of
the vertex set of the Hamming graph . A code for which there
exists an automorphism group that acts transitively on the
code and on its set of neighbours is said to be neighbour transitive, and were
introduced by the authors as a group theoretic analogue to the assumption that
single errors are equally likely over a noisy channel. Examples of neighbour
transitive codes include the Hamming codes, various Golay codes, certain
Hadamard codes, the Nordstrom Robinson codes, certain permutation codes and
frequency permutation arrays, which have connections with powerline
communication, and also completely transitive codes, a subfamily of completely
regular codes, which themselves have attracted a lot of interest. It is known
that for any neighbour transitive code with minimum distance at least 3 there
exists a subgroup of that has a -transitive action on the alphabet over
which the code is defined. Therefore, by Burnside's theorem, this action is of
almost simple or affine type. If the action is of almost simple type, we say
the code is alphabet almost simple neighbour transitive. In this paper we
characterise a family of neighbour transitive codes, in particular, the
alphabet almost simple neighbour transitive codes with minimum distance at
least , and for which the group has a non-trivial intersection with the
base group of . If is such a code, we show that, up to
equivalence, there exists a subcode that can be completely described,
and that either , or is a neighbour transitive frequency
permutation array and is the disjoint union of -translates of .
We also prove that any finite group can be identified in a natural way with a
neighbour transitive code.Comment: 30 Page
Optimality and uniqueness of the Leech lattice among lattices
We prove that the Leech lattice is the unique densest lattice in R^24. The
proof combines human reasoning with computer verification of the properties of
certain explicit polynomials. We furthermore prove that no sphere packing in
R^24 can exceed the Leech lattice's density by a factor of more than
1+1.65*10^(-30), and we give a new proof that E_8 is the unique densest lattice
in R^8.Comment: 39 page
Joint Unitary Triangularization for Gaussian Multi-User MIMO Networks
The problem of transmitting a common message to multiple users over the
Gaussian multiple-input multiple-output broadcast channel is considered, where
each user is equipped with an arbitrary number of antennas. A closed-loop
scenario is assumed, for which a practical capacity-approaching scheme is
developed. By applying judiciously chosen unitary operations at the transmit
and receive nodes, the channel matrices are triangularized so that the
resulting matrices have equal diagonals, up to a possible multiplicative scalar
factor. This, along with the utilization of successive interference
cancellation, reduces the coding and decoding tasks to those of coding and
decoding over the single-antenna additive white Gaussian noise channel. Over
the resulting effective channel, any off-the-shelf code may be used. For the
two-user case, it was recently shown that such joint unitary triangularization
is always possible. In this paper, it is shown that for more than two users, it
is necessary to carry out the unitary linear processing jointly over multiple
channel uses, i.e., space-time processing is employed. It is further shown that
exact triangularization, where all resulting diagonals are equal, is still not
always possible, and appropriate conditions for the existence of such are
established for certain cases. When exact triangularization is not possible, an
asymptotic construction is proposed, that achieves the desired property of
equal diagonals up to edge effects that can be made arbitrarily small, at the
price of processing a sufficiently large number of channel uses together.Comment: Extended version of published paper in IEEE Transactions on
Information Theory, vol. 61, no. 5, pp. 2662-2692, May 201
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