550 research outputs found
A linear construction for certain Kerdock and Preparata codes
The Nordstrom-Robinson, Kerdock, and (slightly modified) Pre\- parata codes
are shown to be linear over \ZZ_4, the integers . The Kerdock and
Preparata codes are duals over \ZZ_4, and the Nordstrom-Robinson code is
self-dual. All these codes are just extended cyclic codes over \ZZ_4. This
provides a simple definition for these codes and explains why their Hamming
weight distributions are dual to each other. First- and second-order
Reed-Muller codes are also linear codes over \ZZ_4, but Hamming codes in
general are not, nor is the Golay code.Comment: 5 page
Embedded Rank Distance Codes for ISI channels
Designs for transmit alphabet constrained space-time codes naturally lead to
questions about the design of rank distance codes. Recently, diversity embedded
multi-level space-time codes for flat fading channels have been designed from
sets of binary matrices with rank distance guarantees over the binary field by
mapping them onto QAM and PSK constellations. In this paper we demonstrate that
diversity embedded space-time codes for fading Inter-Symbol Interference (ISI)
channels can be designed with provable rank distance guarantees. As a corollary
we obtain an asymptotic characterization of the fixed transmit alphabet
rate-diversity trade-off for multiple antenna fading ISI channels. The key idea
is to construct and analyze properties of binary matrices with a particular
structure induced by ISI channels.Comment: Submitted to IEEE Transactions on Information Theor
List decoding of noisy Reed-Muller-like codes
First- and second-order Reed-Muller (RM(1) and RM(2), respectively) codes are
two fundamental error-correcting codes which arise in communication as well as
in probabilistically-checkable proofs and learning. In this paper, we take the
first steps toward extending the quick randomized decoding tools of RM(1) into
the realm of quadratic binary and, equivalently, Z_4 codes. Our main
algorithmic result is an extension of the RM(1) techniques from Goldreich-Levin
and Kushilevitz-Mansour algorithms to the Hankel code, a code between RM(1) and
RM(2). That is, given signal s of length N, we find a list that is a superset
of all Hankel codewords phi with dot product to s at least (1/sqrt(k)) times
the norm of s, in time polynomial in k and log(N). We also give a new and
simple formulation of a known Kerdock code as a subcode of the Hankel code. As
a corollary, we can list-decode Kerdock, too. Also, we get a quick algorithm
for finding a sparse Kerdock approximation. That is, for k small compared with
1/sqrt{N} and for epsilon > 0, we find, in time polynomial in (k
log(N)/epsilon), a k-Kerdock-term approximation s~ to s with Euclidean error at
most the factor (1+epsilon+O(k^2/sqrt{N})) times that of the best such
approximation
Quantum Reed-Solomon Codes
After a brief introduction to both quantum computation and quantum error
correction, we show how to construct quantum error-correcting codes based on
classical BCH codes. With these codes, decoding can exploit additional
information about the position of errors. This error model - the quantum
erasure channel - is discussed. Finally, parameters of quantum BCH codes are
provided.Comment: Summary only (2 pages), for the full version see: Proceedings Applied
Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC-13), Lecture
Notes in Computer Science 1719, Springer, 199
A Theory of Fault-Tolerant Quantum Computation
In order to use quantum error-correcting codes to actually improve the
performance of a quantum computer, it is necessary to be able to perform
operations fault-tolerantly on encoded states. I present a general theory of
fault-tolerant operations based on symmetries of the code stabilizer. This
allows a straightforward determination of which operations can be performed
fault-tolerantly on a given code. I demonstrate that fault-tolerant universal
computation is possible for any stabilizer code. I discuss a number of examples
in more detail, including the five-qubit code.Comment: 30 pages, REVTeX, universal swapping operation added to allow
universal computation on any stabilizer cod
Nonintersecting Subspaces Based on Finite Alphabets
Two subspaces of a vector space are here called ``nonintersecting'' if they
meet only in the zero vector. The following problem arises in the design of
noncoherent multiple-antenna communications systems. How many pairwise
nonintersecting M_t-dimensional subspaces of an m-dimensional vector space V
over a field F can be found, if the generator matrices for the subspaces may
contain only symbols from a given finite alphabet A subseteq F? The most
important case is when F is the field of complex numbers C; then M_t is the
number of antennas. If A = F = GF(q) it is shown that the number of
nonintersecting subspaces is at most (q^m-1)/(q^{M_t}-1), and that this bound
can be attained if and only if m is divisible by M_t. Furthermore these
subspaces remain nonintersecting when ``lifted'' to the complex field. Thus the
finite field case is essentially completely solved. In the case when F = C only
the case M_t=2 is considered. It is shown that if A is a PSK-configuration,
consisting of the 2^r complex roots of unity, the number of nonintersecting
planes is at least 2^{r(m-2)} and at most 2^{r(m-1)-1} (the lower bound may in
fact be the best that can be achieved).Comment: 14 page
Quantum Error Correction and Orthogonal Geometry
A group theoretic framework is introduced that simplifies the description of
known quantum error-correcting codes and greatly facilitates the construction
of new examples. Codes are given which map 3 qubits to 8 qubits correcting 1
error, 4 to 10 qubits correcting 1 error, 1 to 13 qubits correcting 2 errors,
and 1 to 29 qubits correcting 5 errors.Comment: RevTex, 4 pages, no figures, submitted to Phys. Rev. Letters. We have
changed the statement of Theorem 2 to correct it -- we now get worse rates
than we previously claimed for our quantum codes. Minor changes have been
made to the rest of the pape
Scalability of Shor's algorithm with a limited set of rotation gates
Typical circuit implementations of Shor's algorithm involve controlled
rotation gates of magnitude where is the binary length of the
integer N to be factored. Such gates cannot be implemented exactly using
existing fault-tolerant techniques. Approximating a given controlled
rotation gate to within currently requires both
a number of qubits and number of fault-tolerant gates that grows polynomially
with . In this paper we show that this additional growth in space and time
complexity would severely limit the applicability of Shor's algorithm to large
integers. Consequently, we study in detail the effect of using only controlled
rotation gates with less than or equal to some . It is found
that integers up to length can be factored
without significant performance penalty implying that the cumbersome techniques
of fault-tolerant computation only need to be used to create controlled
rotation gates of magnitude if integers thousands of bits long are
desired factored. Explicit fault-tolerant constructions of such gates are also
discussed.Comment: Substantially revised version, twice as long as original. Two tables
converted into one 8-part figure, new section added on the construction of
arbitrary single-qubit rotations using only the fault-tolerant gate set.
Substantial additional discussion and explanatory figures added throughout.
(8 pages, 6 figures
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