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Advances in Mathematics of Communications

2015 , Volume 9 , Issue 1

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Existence conditions for self-orthogonal negacyclic codes over finite fields
Liren Lin, Hongwei Liu and Bocong Chen
2015, 9(1): 1-7 doi: 10.3934/amc.2015.9.1 +[Abstract](359) +[PDF](316.9KB)
Abstract:
In this paper, we obtain necessary and sufficient conditions for the nonexistence of nonzero self-orthogonal negacyclic codes over a finite field, of length relatively prime to the characteristic of the underlying field.
Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties
Zhenyu Zhang, Lijia Ge, Fanxin Zeng and Guixin Xuan
2015, 9(1): 9-21 doi: 10.3934/amc.2015.9.9 +[Abstract](441) +[PDF](424.1KB)
Abstract:
In this paper, a novel method for constructing complementary sequence set with zero correlation zone (ZCZ) is presented by interleaving and combining three orthogonal matrices. The constructed set can be divided into multiple sequence groups and each sequence group can be further divided into multiple sequence subgroups. In addition to ZCZ properties of sequences from the same sequence subgroup, sequences from different sequence groups are orthogonal to each other while sequences from different sequence subgroups within the same sequence group possess ideal cross-correlation properties, that is, the proposed ZCZ sequence set has inter-group orthogonal (IGO) and inter-subgroup complementary (ISC) properties. Compared with previous methods, the new construction can provide flexible choice for ZCZ width and set size, and the resultant sequences which are called IGO-ISC sequences in this paper can achieve the theoretical bound on the set size for the ZCZ width and sequence length.
Some new classes of cyclic codes with three or six weights
Yongbo Xia, Tor Helleseth and Chunlei Li
2015, 9(1): 23-36 doi: 10.3934/amc.2015.9.23 +[Abstract](360) +[PDF](445.7KB)
Abstract:
In this paper, a class of three-weight cyclic codes over prime fields $\mathbb{F}_p$ of odd order whose duals have two zeros, and a class of six-weight cyclic codes whose duals have three zeros are presented. The weight distributions of these cyclic codes are derived.
Plaintext checkable encryption with designated checker
Angsuman Das, Avishek Adhikari and Kouichi Sakurai
2015, 9(1): 37-53 doi: 10.3934/amc.2015.9.37 +[Abstract](260) +[PDF](415.0KB)
Abstract:
This paper introduces a new public-key primitive called designated plaintext checkable encryption (DPCE) in which given a ciphertext, a delegated checker can determine whether the ciphertext decrypts under the same public key to a plaintext chosen by himself. Motivated by various applications, two types of DPCE (of Type-I and II) are defined, depending upon whether the user delegates the plaintext checking right at his will to a delegated checker (Type-I) or the user is required to provide this plaintext checking right to a designated checker (Type-II). We propose several generic random-oracle and standard model constructions for DPCE of both the types based on arbitrary probabilistic or deterministic encryption schemes.
A lower bound on the average Hamming correlation of frequency-hopping sequence sets
Aixian Zhang, Zhengchun Zhou and Keqin Feng
2015, 9(1): 55-62 doi: 10.3934/amc.2015.9.55 +[Abstract](338) +[PDF](294.3KB)
Abstract:
The average Hamming correlation is an important indicator of frequency-hopping sequences (FHSs) which measures the average performance of FHSs employed in practical frequency-hopping multiple access (FHMA) communication systems. In this paper, a lower bound on average Hamming auto- and cross correlations of an FHS set is derived. It generalizes and improves the lower bound proposed recently by Peng, Niu and Tang. A simple necessary and sufficient condition for an FHS set to meet the new bound is given. Based on this condition, two classes of FHS sets whose average Hamming correlations reach the proposed bound are introduced.
Multiple coverings of the farthest-off points with small density from projective geometry
Daniele Bartoli, Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini and Fernanda Pambianco
2015, 9(1): 63-85 doi: 10.3934/amc.2015.9.63 +[Abstract](321) +[PDF](506.9KB)
Abstract:
In this paper we deal with the special class of covering codes consisting of multiple coverings of the farthest-off points (MCF). In order to measure the quality of an MCF code, we use a natural extension of the notion of density for ordinary covering codes, that is the $\mu$-density for MCF codes; a generalization of the length function for linear covering codes is also introduced. Our main results consist in a number of upper bounds on such a length function, obtained through explicit constructions, especially for the case of covering radius $R=2$. A key tool is the possibility of computing the $\mu$-length function in terms of Projective Geometry over finite fields. In fact, linear $(R,\mu )$-MCF codes with parameters $ [n,n-r,d]_{q}R$ have a geometrical counterpart consisting of special subsets of $n$ points in the projective space $PG(n-r-1,q)$. We introduce such objects under the name of $(\rho,\mu)$-saturating sets and we provide a number of example and existence results. Finally, Almost Perfect MCF (APMCF) codes, that is codes for which each word at distance $R$ from the code belongs to {exactly} $\mu $ spheres centered in codewords, are considered and their connections with uniformly packed codes, two-weight codes, and subgroups of Singer groups are pointed out.
Polar codes for distributed hierarchical source coding
Min Ye and Alexander Barg
2015, 9(1): 87-103 doi: 10.3934/amc.2015.9.87 +[Abstract](867) +[PDF](440.0KB)
Abstract:
We show that polar codes can be used to achieve the rate-distortion functions in the problem of hierarchical source coding also known as the successive refinement problem. We also analyze the distributed version of this problem, constructing a polar coding scheme that achieves the rate distortion functions for successive refinement with side information.
Derived and residual subspace designs
Michael Kiermaier and Reinhard Laue
2015, 9(1): 105-115 doi: 10.3934/amc.2015.9.105 +[Abstract](347) +[PDF](372.1KB)
Abstract:
A generalization of forming derived and residual designs from $t$-designs to subspace designs is proposed. A $q$-analog of a theorem by Tran Van Trung, van Leijenhorst and Driessen is proven, stating that if for some (not necessarily realizable) parameter set the derived and residual parameter set are realizable, the same is true for the reduced parameter set.
    As a result, we get the existence of several previously unknown subspace designs. Some consequences are derived for the existence of large sets of subspace designs. Furthermore, it is shown that there is no $q$-analog of the large Witt design.
Two new classes of binary sequence pairs with three-level cross-correlation
Xiaohui Liu, Jinhua Wang and Dianhua Wu
2015, 9(1): 117-128 doi: 10.3934/amc.2015.9.117 +[Abstract](299) +[PDF](357.0KB)
Abstract:
A pair of binary sequences is generalized from the concept of a two-level autocorrelation function of single binary sequence. In this paper, we describe two classes of binary sequence pairs of period $N=2q$, where $q=4f+1$ is an odd prime and $f$ is an even integer. Those classes of binary sequence pairs are based on cyclic almost difference set pairs. They have optimal three-level cross-correlation, and either balanced or almost balanced.

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