7,092 research outputs found
Some New Constructions of Quantum MDS Codes
It is an important task to construct quantum maximum-distance-separable (MDS)
codes with good parameters. In the present paper, we provide six new classes of
q-ary quantum MDS codes by using generalized Reed-Solomon (GRS) codes and
Hermitian construction. The minimum distances of our quantum MDS codes can be
larger than q/2+1 Three of these six classes of quantum MDS codes have longer
lengths than the ones constructed in [1] and [2], hence some of their results
can be easily derived from ours via the propagation rule. Moreover, some known
quantum MDS codes of specific lengths can be seen as special cases of ours and
the minimum distances of some known quantum MDS codes are also improved as
well.Comment: Accepted for publication in IEEE Transactions on Information Theor
Two new classes of quantum MDS codes
Let be a prime and let be a power of . In this paper, by using
generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we
construct two new classes of quantum maximum-distance- separable (MDS) codes
with parameters for any , and for any with . Our quantum codes have flexible parameters, and have minimum
distances larger than when . Furthermore, it
turns out that our constructions generalize and improve some previous results.Comment: 14 pages. Accepted by Finite Fields and Their Application
Optimal cyclic locally repairable codes with unbounded length
Locally repairable codes with locality (-LRCs for short) were
introduced by Gopalan et al. \cite{1} to recover a failed node of the code from
at most other available nodes. And then locally repairable
codes (-LRCs for short) were produced by Prakash et al. \cite{2}
for tolerating multiple failed nodes. An -LRC can be viewed as an
-LRC. An -LRC is called optimal if it achieves the
Singleton-type bound. It has been a great challenge to construct -ary
optimal -LRCs with length much larger than . Surprisingly, Luo
et al. \cite{3} presented a construction of -ary optimal -LRCs of minimum
distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are
independent of ) via cyclic codes.
In this paper, inspired by the work of \cite{3}, we firstly construct two
classes of optimal cyclic -LRCs with unbounded lengths and minimum
distances or , which generalize the results about the
case given in \cite{3}. Secondly, with a slightly stronger
condition, we present a construction of optimal cyclic -LRCs with
unbounded length and larger minimum distance . Furthermore, when
, we give another class of optimal cyclic -LRCs with unbounded
length and minimum distance
On Random Linear Network Coding for Butterfly Network
Random linear network coding is a feasible encoding tool for network coding,
specially for the non-coherent network, and its performance is important in
theory and application. In this letter, we study the performance of random
linear network coding for the well-known butterfly network by analyzing the
failure probabilities. We determine the failure probabilities of random linear
network coding for the well-known butterfly network and the butterfly network
with channel failure probability p.Comment: This paper was submitted to IEEE Communications Letter
Linear Network Error Correction Multicast/Broadcast/Dispersion/Generic Codes
In the practical network communications, many internal nodes in the network
are required to not only transmit messages but decode source messages. For
different applications, four important classes of linear network codes in
network coding theory, i.e., linear multicast, linear broadcast, linear
dispersion, and generic network codes, have been studied extensively. More
generally, when channels of communication networks are noisy, information
transmission and error correction have to be under consideration
simultaneously, and thus these four classes of linear network codes are
generalized to linear network error correction (LNEC) coding, and we say them
LNEC multicast, broadcast, dispersion, and generic codes, respectively.
Furthermore, in order to characterize their efficiency of information
transmission and error correction, we propose the (weakly, strongly) extended
Singleton bounds for them, and define the corresponding optimal codes, i.e.,
LNEC multicast/broadcast/dispersion/generic MDS codes, which satisfy the
corresponding Singleton bounds with equality. The existences of such MDS codes
are discussed in detail by algebraic methods and the constructive algorithms
are also proposed.Comment: Single column, 38 pages. Submitted for possible publicatio
Johnson Type Bounds on Constant Dimension Codes
Very recently, an operator channel was defined by Koetter and Kschischang
when they studied random network coding. They also introduced constant
dimension codes and demonstrated that these codes can be employed to correct
errors and/or erasures over the operator channel. Constant dimension codes are
equivalent to the so-called linear authentication codes introduced by Wang,
Xing and Safavi-Naini when constructing distributed authentication systems in
2003. In this paper, we study constant dimension codes. It is shown that
Steiner structures are optimal constant dimension codes achieving the
Wang-Xing-Safavi-Naini bound. Furthermore, we show that constant dimension
codes achieve the Wang-Xing-Safavi-Naini bound if and only if they are certain
Steiner structures. Then, we derive two Johnson type upper bounds, say I and
II, on constant dimension codes. The Johnson type bound II slightly improves on
the Wang-Xing-Safavi-Naini bound. Finally, we point out that a family of known
Steiner structures is actually a family of optimal constant dimension codes
achieving both the Johnson type bounds I and II.Comment: 12 pages, submitted to Designs, Codes and Cryptograph
Optical analogy to quantum Fourier transform based on pseudorandom phase ensemble
In this paper, we introduce an optical analogy to quantum Fourier
tanformation based on a pseudorandom phase ensemble. The optical analogy also
brings about exponential speedup over classical Fourier tanformation. Using the
analogy, we demonstrate three classcial fields to realize Fourier transform
similar to three quantum particles.Comment: A small amount of errors modification,16 Pages, 1 figur
Minimum Pseudo-Weight and Minimum Pseudo-Codewords of LDPC Codes
In this correspondence, we study the minimum pseudo-weight and minimum
pseudo-codewords of low-density parity-check (LDPC) codes under linear
programming (LP) decoding. First, we show that the lower bound of Kelly,
Sridhara, Xu and Rosenthal on the pseudo-weight of a pseudo-codeword of an LDPC
code with girth greater than 4 is tight if and only if this pseudo-codeword is
a real multiple of a codeword. Then, we show that the lower bound of Kashyap
and Vardy on the stopping distance of an LDPC code is also a lower bound on the
pseudo-weight of a pseudo-codeword of this LDPC code with girth 4, and this
lower bound is tight if and only if this pseudo-codeword is a real multiple of
a codeword. Using these results we further show that for some LDPC codes, there
are no other minimum pseudo-codewords except the real multiples of minimum
codewords. This means that the LP decoding for these LDPC codes is
asymptotically optimal in the sense that the ratio of the probabilities of
decoding errors of LP decoding and maximum-likelihood decoding approaches to 1
as the signal-to-noise ratio leads to infinity. Finally, some LDPC codes are
listed to illustrate these results.Comment: 17 pages, 1 figur
The minimal polynomial of sequence obtained from componentwise linear transformation of linear recurring sequence
Let be a linear recurring sequence with terms in
and be a linear transformation of over . Denote
. In this paper, we first present counter
examples to show the main result in [A.M. Youssef and G. Gong, On linear
complexity of sequences over , Theoretical Computer Science,
352(2006), 288-292] is not correct in general since Lemma 3 in that paper is
incorrect. Then, we determine the minimal polynomial of if the canonical
factorization of the minimal polynomial of without multiple roots is known
and thus present the solution to the problem which was mainly considered in the
above paper but incorrectly solved. Additionally, as a special case, we
determine the minimal polynomial of if the minimal polynomial of is
primitive. Finally, we give an upper bound on the linear complexity of
when exhausts all possible linear transformations of over
. This bound is tight in some cases.Comment: This paper was submitted to the journal Theoretical Computer Scienc
The Minimal Polynomial over F_q of Linear Recurring Sequence over F_{q^m}
Recently, motivated by the study of vectorized stream cipher systems, the
joint linear complexity and joint minimal polynomial of multisequences have
been investigated. Let S be a linear recurring sequence over finite field
F_{q^m} with minimal polynomial h(x) over F_{q^m}. Since F_{q^m} and F_{q}^m
are isomorphic vector spaces over the finite field F_q, S is identified with an
m-fold multisequence S^{(m)} over the finite field F_q. The joint minimal
polynomial and joint linear complexity of the m-fold multisequence S^{(m)} are
the minimal polynomial and linear complexity over F_q of S respectively. In
this paper, we study the minimal polynomial and linear complexity over F_q of a
linear recurring sequence S over F_{q^m} with minimal polynomial h(x) over
F_{q^m}. If the canonical factorization of h(x) in F_{q^m}[x] is known, we
determine the minimal polynomial and linear complexity over F_q of the linear
recurring sequence S over F_{q^m}.Comment: Submitted to the journal Finite Fields and Their Application
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