Nonexistence of Generalized Bent Functions From Z2nZ_{2}^{n} to ZmZ_{m}

Several nonexistence results on generalized bent functions $f:Z_{2}^{n} \rightarrow Z_{m}$ 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 $\mathbb{F}_{q^n} / \mathbb{F}_{q}$ 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 $q=2^n$ 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 $f$, 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 $f$. 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 $f$

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 $p$ 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 $p$ 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 $k$-uniform if all its reductions to $k$-qudit are maximally mixed. We investigate the general constructions of $k$-uniform pure quantum states of $n$ subsystems with $d$ 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 $n\ge 2$, there always exists an $n/2$-uniform $n$-qudit quantum state of level $p$ for sufficiently large prime $p$. Secondly, both constructions show that their exist $k$-uniform $n$-qudit pure quantum states such that $k$ is proportional to $n$, i.e., $k=\Omega(n)$ 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 $3$-uniform $n$-qudit pure quantum state for all $n\ge 8$. In fact, we can further prove that, for every $k$, there exists a constant $M_k$ such that there exists a $k$-uniform $n$-qudit quantum state for all $n\ge M_k$. In addition, by using concatenation of algebraic geometry codes, we give an explicit construction of $k$-uniform quantum state when $k$ tends to infinity