8,672 research outputs found
Improved constructions of nested code pairs
Producción CientíficaTwo new constructions of linear nested code pairs are given for which the codimension and the relative minimum distances of the codes and their duals are good. By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large. Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes. They can also be used to ensure reliable communication over asymmetric quantum channels. The new constructions result from carefully applying the Feng-Rao bounds to a family of codes defined from multivariate polynomials and Cartesian product point sets.The Danish Council for Independent Research (Grant N.
DFF–4002-00367)Ministerio de Economía, Industria y Competitividad (Projects MTM2015-65764-C3-2-P and MTM2015-69138-REDT)University Jaume I (Grant N. P1-1B2015-02
On nested code pairs from the Hermitian curve
Nested code pairs play a crucial role in the construction of ramp secret
sharing schemes [Kurihara et al. 2012] and in the CSS construction of quantum
codes [Ketkar et al. 2006]. The important parameters are (1) the codimension,
(2) the relative minimum distance of the codes, and (3) the relative minimum
distance of the dual set of codes. Given values for two of them, one aims at
finding a set of nested codes having parameters with these values and with the
remaining parameter being as large as possible. In this work we study nested
codes from the Hermitian curve. For not too small codimension, we present
improved constructions and provide closed formula estimates on their
performance. For small codimension we show how to choose pairs of one-point
algebraic geometric codes in such a way that one of the relative minimum
distances is larger than the corresponding non-relative minimum distance.Comment: 28 page
Steane-Enlargement of Quantum Codes from the Hermitian Curve
In this paper, we study the construction of quantum codes by applying
Steane-enlargement to codes from the Hermitian curve. We cover
Steane-enlargement of both usual one-point Hermitian codes and of order bound
improved Hermitian codes. In particular, the paper contains two constructions
of quantum codes whose parameters are described by explicit formulae, and we
show that these codes compare favourably to existing, comparable constructions
in the literature.Comment: 11 page
On Steane-Enlargement of Quantum Codes from Cartesian Product Point Sets
In this work, we study quantum error-correcting codes obtained by using
Steane-enlargement. We apply this technique to certain codes defined from
Cartesian products previously considered by Galindo et al. in [4]. We give
bounds on the dimension increase obtained via enlargement, and additionally
give an algorithm to compute the true increase. A number of examples of codes
are provided, and their parameters are compared to relevant codes in the
literature, which shows that the parameters of the enlarged codes are
advantageous. Furthermore, comparison with the Gilbert-Varshamov bound for
stabilizer quantum codes shows that several of the enlarged codes match or
exceed the parameters promised by the bound.Comment: 12 page
Secure Compute-and-Forward in a Bidirectional Relay
We consider the basic bidirectional relaying problem, in which two users in a
wireless network wish to exchange messages through an intermediate relay node.
In the compute-and-forward strategy, the relay computes a function of the two
messages using the naturally-occurring sum of symbols simultaneously
transmitted by user nodes in a Gaussian multiple access (MAC) channel, and the
computed function value is forwarded to the user nodes in an ensuing broadcast
phase. In this paper, we study the problem under an additional security
constraint, which requires that each user's message be kept secure from the
relay. We consider two types of security constraints: perfect secrecy, in which
the MAC channel output seen by the relay is independent of each user's message;
and strong secrecy, which is a form of asymptotic independence. We propose a
coding scheme based on nested lattices, the main feature of which is that given
a pair of nested lattices that satisfy certain "goodness" properties, we can
explicitly specify probability distributions for randomization at the encoders
to achieve the desired security criteria. In particular, our coding scheme
guarantees perfect or strong secrecy even in the absence of channel noise. The
noise in the channel only affects reliability of computation at the relay, and
for Gaussian noise, we derive achievable rates for reliable and secure
computation. We also present an application of our methods to the multi-hop
line network in which a source needs to transmit messages to a destination
through a series of intermediate relays.Comment: v1 is a much expanded and updated version of arXiv:1204.6350; v2 is a
minor revision to fix some notational issues; v3 is a much expanded and
updated version of v2, and contains results on both perfect secrecy and
strong secrecy; v3 is a revised manuscript submitted to the IEEE Transactions
on Information Theory in April 201
Constructions of Pure Asymmetric Quantum Alternant Codes Based on Subclasses of Alternant Codes
In this paper, we construct asymmetric quantum error-correcting codes(AQCs)
based on subclasses of Alternant codes. Firstly, We propose a new subclass of
Alternant codes which can attain the classical Gilbert-Varshamov bound to
construct AQCs. It is shown that when , -parts of the AQCs can attain
the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous
subclass of Alternant codes called Goppa codes. As an illustrative example, we
get three AQCs from the well
known binary Goppa code. At last, we get asymptotically good
binary expansions of asymmetric quantum GRS codes, which are quantum
generalizations of Retter's classical results. All the AQCs constructed in this
paper are pure
- …