143 research outputs found
Nonexistence of Generalized Bent Functions From to
Several nonexistence results on generalized bent functions presented by using some knowledge on cyclotomic number
fields and their imaginary quadratic subfields
Relative Generalized Hamming Weights of Cyclic Codes
Relative generalized Hamming weights (RGHWs) of a linear code respect to a
linear subcode determine the security of the linear ramp secret sharing scheme
based on the code. They can be used to express the information leakage of the
secret when some keepers of shares are corrupted. Cyclic codes are an
interesting type of linear codes and have wide applications in communication
and storage systems. In this paper, we investigate the RGHWs of cyclic codes
with two nonzeros respect to any of its irreducible cyclic subcodes. Applying
the method in the paper [arxiv.org/abs/1410.2702], we give two formulae for
RGHWs of the cyclic codes. As applications of the formulae, explicit examples
are computed.Comment: 14 page
A New Criterion on Normal Bases of Finite Field Extensions
A new criterion on normal bases of finite field extension is presented and explicit criterions for several particular
finite field extensions are derived from this new criterion
An Unified Approach on Constructing of MDS Self-dual Codes via Reed-Solomon Codes
Based on the fundamental results on MDS self-dual codes over finite fields
constructed via generalized Reed-Solomon codes \cite{JX} and extended
generalized Reed-Solomon codes \cite{Yan}, many series of MDS self-dual codes
with different length have been obtained recently by a variety of constructions
and individual computations. In this paper, we present an unified approach to
get several previous results with concise statements and simplified proofs, and
some new constructions on MDS self-dual codes. In the conclusion section we
raise two open problems.Comment: 16page
A Construction of Linear Codes over \f_{2^t} from Boolean Functions
In this paper, we present a construction of linear codes over \f_{2^t} from
Boolean functions, which is a generalization of Ding's method \cite[Theorem
9]{Ding15}. Based on this construction, we give two classes of linear codes
\tilde{\C}_{f} and \C_f (see Theorem \ref{thm-maincode1} and Theorem
\ref{thm-maincodenew}) over \f_{2^t} from a Boolean function
f:\f_{q}\rightarrow \f_2, where and \f_{2^t} is some subfield of
\f_{q}. The complete weight enumerator of \tilde{\C}_{f} can be easily
determined from the Walsh spectrum of , while the weight distribution of the
code \C_f can also be easily settled. Particularly, the number of nonzero
weights of \tilde{\C}_{f} and \C_f is the same as the number of distinct
Walsh values of . As applications of this construction, we show several
series of linear codes over \f_{2^t} with two or three weights by using bent,
semibent, monomial and quadratic Boolean function
Construction of Cyclic and Constacyclic Codes for b-symbol Read Channels Meeting the Plotkin-like Bound
The symbol-pair codes over finite fields have been raised for symbol-pair
read channels and motivated by application of high-density data storage
technologies [1, 2]. Their generalization is the code for b-symbol read
channels (b > 2). Many MDS codes for b-symbol read channels have been
constructed which meet the Singleton-like bound ([3, 4, 10] for b = 2 and [11]
for b > 2). In this paper we show the Plotkin-like bound and present a
construction on irreducible cyclic codes and constacyclic codes meeting the
Plotkin-like bound
A Class of Linear Codes with a Few Weights
Linear codes have been an interesting subject of study for many years, as
linear codes with few weights have applications in secrete sharing,
authentication codes, association schemes, and strongly regular graphs. In this
paper, a class of linear codes with a few weights over the finite field
\gf(p) are presented and their weight distributions are also determined,
where is an odd prime. Some of the linear codes obtained are optimal in the
sense that they meet certain bounds on linear codes.Comment: arXiv admin note: text overlap with arXiv:1503.06512 by other author
Linear codes with a few weights from inhomogeneous quadratic functions
Linear codes with few weights have been an interesting subject of study for
many years, as these codes have applications in secrete sharing, authentication
codes, association schemes, and strongly regular graphs. In this paper, linear
codes with a few weights are constructed from inhomogeneous quadratic functions
over the finite field \gf(p), where is an odd prime. They include some
earlier linear codes as special cases. The weight distributions of these linear
codes are also determined
Multivariate Interpolation Formula over Finite Fields and Its Applications in Coding Theory
A multivariate interpolation formula (MVIF) over finite fields is presented
by using the proposed Kronecker delta function. The MVIF can be applied to
yield polynomial relations over the base field among homogeneous symmetric
rational functions. Besides the property that all the coefficients are coming
from the base field, there is also a significant one on the degrees of the
obtained polynomial; namely, the degree of each term satisfies certain
condition. Next, for any cyclic codes the unknown syndrome representation can
also be provided by the proposed MVIF and also has the same properties. By
applying the unknown syndrome representation and the Berlekamp-Massey
algorithm, one-step decoding algorithms can be developed to determine the error
locator polynomials for arbitrary cyclic codes.Comment: 11 pages. This work is supported by the grant of the NSFC no.10990011
and the Tsinghua National Lab. of Information Science and Technolog
Multipartite entangled states, symmetric matrices and error-correcting codes
A pure quantum state is called -uniform if all its reductions to -qudit
are maximally mixed. We investigate the general constructions of -uniform
pure quantum states of subsystems with levels. We provide one
construction via symmetric matrices and the second one through classical
error-correcting codes. There are three main results arising from our
constructions. Firstly, we show that for any given even , there always
exists an -uniform -qudit quantum state of level for sufficiently
large prime . Secondly, both constructions show that their exist -uniform
-qudit pure quantum states such that is proportional to , i.e.,
although the construction from symmetric matrices outperforms the
one by error-correcting codes. Thirdly, our symmetric matrix construction
provides a positive answer to the open question in \cite{DA} on whether there
exists -uniform -qudit pure quantum state for all . In fact, we
can further prove that, for every , there exists a constant such that
there exists a -uniform -qudit quantum state for all . In
addition, by using concatenation of algebraic geometry codes, we give an
explicit construction of -uniform quantum state when tends to infinity
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