ISSN:
1930-5346
eISSN:
1930-5338
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Advances in Mathematics of Communications
May 2014 , Volume 8 , Issue 2
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2014, 8(2): 119-127
doi: 10.3934/amc.2014.8.119
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Abstract:
A problem of improving the accuracy of nonparametric entropy estimation for a stationary ergodic process is considered. New weak metrics are introduced and relations between metrics, measures, and entropy are discussed. A new nonparametric entropy estimator is constructed based on weak metrics and has a parameter with which the estimator is optimized to reduce its bias. It is shown that estimator's variance is upper-bounded by a nearly optimal Cramér-Rao lower bound.
A problem of improving the accuracy of nonparametric entropy estimation for a stationary ergodic process is considered. New weak metrics are introduced and relations between metrics, measures, and entropy are discussed. A new nonparametric entropy estimator is constructed based on weak metrics and has a parameter with which the estimator is optimized to reduce its bias. It is shown that estimator's variance is upper-bounded by a nearly optimal Cramér-Rao lower bound.
2014, 8(2): 129-137
doi: 10.3934/amc.2014.8.129
+[Abstract](1764)
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Abstract:
This paper gives lower and upper bounds on the covering radius of codes over $\mathbb{Z}_{2^s}$ with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type $\alpha$ and Type $\beta$) and their dual and give bounds on the covering radii for MacDonald codes of both types over $\mathbb{Z}_4$.
This paper gives lower and upper bounds on the covering radius of codes over $\mathbb{Z}_{2^s}$ with respect to homogenous distance. We also determine the covering radius of various Repetition codes, Simplex codes (Type $\alpha$ and Type $\beta$) and their dual and give bounds on the covering radii for MacDonald codes of both types over $\mathbb{Z}_4$.
2014, 8(2): 139-152
doi: 10.3934/amc.2014.8.139
+[Abstract](1716)
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Abstract:
A set of quasi-uniform random variables $X_1,\ldots,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in more detail different families of non-abelian groups with respect to the entropic vectors they yield. In particular, we address the question of whether a given non-abelian group $G$ and some fixed subgroups $G_1,\ldots,G_n$ end up giving the same entropic vector as some abelian group $A$ with subgroups $A_1,\ldots,A_n$, in which case we say that $(A, A_1, \ldots, A_n)$ represents $(G, G_1, \ldots, G_n)$. If for any choice of subgroups $G_1,\ldots,G_n$, there exists some abelian group $A$ which represents $G$, we refer to $G$ as being abelian (group) representable for $n$. We completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when $n=2$, for which we show a group is abelian representable if and only if it is nilpotent. This problem is motivated by understanding non-linear coding strategies for network coding, and network information theory capacity regions.
A set of quasi-uniform random variables $X_1,\ldots,X_n$ may be generated from a finite group $G$ and $n$ of its subgroups, with the corresponding entropic vector depending on the subgroup structure of $G$. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in more detail different families of non-abelian groups with respect to the entropic vectors they yield. In particular, we address the question of whether a given non-abelian group $G$ and some fixed subgroups $G_1,\ldots,G_n$ end up giving the same entropic vector as some abelian group $A$ with subgroups $A_1,\ldots,A_n$, in which case we say that $(A, A_1, \ldots, A_n)$ represents $(G, G_1, \ldots, G_n)$. If for any choice of subgroups $G_1,\ldots,G_n$, there exists some abelian group $A$ which represents $G$, we refer to $G$ as being abelian (group) representable for $n$. We completely characterize dihedral, quasi-dihedral and dicyclic groups with respect to their abelian representability, as well as the case when $n=2$, for which we show a group is abelian representable if and only if it is nilpotent. This problem is motivated by understanding non-linear coding strategies for network coding, and network information theory capacity regions.
2014, 8(2): 153-165
doi: 10.3934/amc.2014.8.153
+[Abstract](1721)
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Abstract:
To resist Binary Decision Diagrams (BDD) based attacks, a Boolean function should have a high BDD size. The hidden weighted bit function (HWBF), introduced by Bryant in 1991, seems to be the simplest function with exponential BDD size. In [28], Wang et al. investigated the cryptographic properties of the HWBF and found that it is a very good candidate for being used in real ciphers. In this paper, we modify the HWBF and construct two classes of functions with very good cryptographic properties (better than the HWBF). The new functions are balanced, with almost optimum algebraic degree and satisfy the strict avalanche criterion. Their nonlinearity is higher than that of the HWBF. We investigate their algebraic immunity, BDD size and their resistance against fast algebraic attacks, which seem to be better than those of the HWBF too. The new functions are simple, can be implemented efficiently, have high BDD sizes and rather good cryptographic properties. Therefore, they might be excellent candidates for constructions of real-life ciphers.
To resist Binary Decision Diagrams (BDD) based attacks, a Boolean function should have a high BDD size. The hidden weighted bit function (HWBF), introduced by Bryant in 1991, seems to be the simplest function with exponential BDD size. In [28], Wang et al. investigated the cryptographic properties of the HWBF and found that it is a very good candidate for being used in real ciphers. In this paper, we modify the HWBF and construct two classes of functions with very good cryptographic properties (better than the HWBF). The new functions are balanced, with almost optimum algebraic degree and satisfy the strict avalanche criterion. Their nonlinearity is higher than that of the HWBF. We investigate their algebraic immunity, BDD size and their resistance against fast algebraic attacks, which seem to be better than those of the HWBF too. The new functions are simple, can be implemented efficiently, have high BDD sizes and rather good cryptographic properties. Therefore, they might be excellent candidates for constructions of real-life ciphers.
2014, 8(2): 167-189
doi: 10.3934/amc.2014.8.167
+[Abstract](1982)
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Abstract:
In this paper, we focus on the design of unitary space-time codes achieving full diversity using division algebras, and on the systematic computation of their minimum determinant. We also give examples of such codes with high minimum determinant. Division algebras allow to obtain higher rates than known constructions based on finite groups.
In this paper, we focus on the design of unitary space-time codes achieving full diversity using division algebras, and on the systematic computation of their minimum determinant. We also give examples of such codes with high minimum determinant. Division algebras allow to obtain higher rates than known constructions based on finite groups.
2014, 8(2): 191-207
doi: 10.3934/amc.2014.8.191
+[Abstract](1595)
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Abstract:
The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under character-theoretic dualization. A characterization is given of those rings for which the induced partition is reflexive or even self-dual.
The values of the homogeneous weight are determined for finite Frobenius rings that are a direct product of local Frobenius rings. This is used to investigate the partition induced by this weight and its dual partition under character-theoretic dualization. A characterization is given of those rings for which the induced partition is reflexive or even self-dual.
2014, 8(2): 209-222
doi: 10.3934/amc.2014.8.209
+[Abstract](1623)
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Abstract:
A binary sequence family ${\mathcal S}$ of length $n$ and size $M$ can be characterized by the maximum magnitude of its nontrivial aperiodic correlation, denoted as $\theta_{\max} ({\mathcal S})$. The lower bound on $\theta_{\max} ({\mathcal S})$ was originally presented by Welch, and improved later by Levenshtein. In this paper, a Fourier transform approach is introduced in an attempt to improve the Levenshtein's lower bound. Through the approach, a new expression of the Levenshtein bound is developed. Along with numerical supports, it is found that $\theta_{\max} ^2 ({\mathcal S}) > 0.3584 n-0.0810$ for $M=3$ and $n \ge 4$, and $\theta_{\max} ^2 ({\mathcal S}) > 0.4401 n-0.1053$ for $M=4$ and $n \ge 4$, respectively, which are tighter than the original Welch and Levenshtein bounds.
A binary sequence family ${\mathcal S}$ of length $n$ and size $M$ can be characterized by the maximum magnitude of its nontrivial aperiodic correlation, denoted as $\theta_{\max} ({\mathcal S})$. The lower bound on $\theta_{\max} ({\mathcal S})$ was originally presented by Welch, and improved later by Levenshtein. In this paper, a Fourier transform approach is introduced in an attempt to improve the Levenshtein's lower bound. Through the approach, a new expression of the Levenshtein bound is developed. Along with numerical supports, it is found that $\theta_{\max} ^2 ({\mathcal S}) > 0.3584 n-0.0810$ for $M=3$ and $n \ge 4$, and $\theta_{\max} ^2 ({\mathcal S}) > 0.4401 n-0.1053$ for $M=4$ and $n \ge 4$, respectively, which are tighter than the original Welch and Levenshtein bounds.
2014, 8(2): 223-239
doi: 10.3934/amc.2014.8.223
+[Abstract](1609)
+[PDF](4404.8KB)
Abstract:
In the present article we propose a reduction point algorithm for any Fuchsian group in the absence of parabolic transformations. We extend to this setting classical algorithms for Fuchsian groups with parabolic transformations, such as the flip flop algorithm known for the modular group $\mathbf{SL}(2, \mathbb{Z})$ and whose roots go back to [9]. The research has been partially motivated by the need to design more efficient codes for wireless transmission data and for the study of Maass waveforms under a computational point of view.
In the present article we propose a reduction point algorithm for any Fuchsian group in the absence of parabolic transformations. We extend to this setting classical algorithms for Fuchsian groups with parabolic transformations, such as the flip flop algorithm known for the modular group $\mathbf{SL}(2, \mathbb{Z})$ and whose roots go back to [9]. The research has been partially motivated by the need to design more efficient codes for wireless transmission data and for the study of Maass waveforms under a computational point of view.
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