ISSN:
1930-5346
eISSN:
1930-5338
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Advances in Mathematics of Communications
February 2008 , Volume 2 , Issue 1
The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has influenced dynamical systems as deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other fields of core mathematics. For the full preface, please click the "full text" button above. Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson, and Anton Zorich.
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2008, 2(1): 1-13
doi: 10.3934/amc.2008.2.1
+[Abstract](2088)
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Abstract:
The problem of the estimation of the entropy rate of a stationary ergodic process $\mu$ is considered. A new nonparametric entropy rate estimator is constructed for a sample of n sequences $(X_1$(1)$,\ldots, (X_m$(1)$),\ldots, (X_1$(n) $,\ldots, (X_m$(n)$)$ independently generated by $\mu$. It is shown that, for $m=O(\log n)$, the estimator converges almost surely and its variance is upper-bounded by $O(n$−1$)$ for a large class of stationary ergodic processes with a finite state space. As the order $O(n$−1$)$ of the variance growth on $n$ is the same as that of the optimal Cramer-Rao lower bound, presented is the first near-optimal estimator in the sense of the variance convergence.
The problem of the estimation of the entropy rate of a stationary ergodic process $\mu$ is considered. A new nonparametric entropy rate estimator is constructed for a sample of n sequences $(X_1$(1)$,\ldots, (X_m$(1)$),\ldots, (X_1$(n) $,\ldots, (X_m$(n)$)$ independently generated by $\mu$. It is shown that, for $m=O(\log n)$, the estimator converges almost surely and its variance is upper-bounded by $O(n$−1$)$ for a large class of stationary ergodic processes with a finite state space. As the order $O(n$−1$)$ of the variance growth on $n$ is the same as that of the optimal Cramer-Rao lower bound, presented is the first near-optimal estimator in the sense of the variance convergence.
2008, 2(1): 15-33
doi: 10.3934/amc.2008.2.15
+[Abstract](2065)
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Abstract:
Improvements to code dimension of evaluation codes, while maintaining a fixed decoding radius, were discovered by Feng and Rao, 1995, and nicely described in terms of an order function by Høholdt, van Lint, Pellikaan, 1998. In an earlier work, 2006, we considered a different improvement, based on the observation that the decoding algorithm corrects an error vector based not so much on the weight of the vector but rather the ''footprint'' of the error locations. In both cases one can find minimal sets of parity checks defining the codes by means of the order function. In this paper we show that these minimal sets have a very useful closure property. For several important families of codes that we consider, this property allows us to construct a generating matrix for the code that has properties amenable to encoding. The generating matrix can be constructed by evaluating monomials in a set which also has the closure property.
Improvements to code dimension of evaluation codes, while maintaining a fixed decoding radius, were discovered by Feng and Rao, 1995, and nicely described in terms of an order function by Høholdt, van Lint, Pellikaan, 1998. In an earlier work, 2006, we considered a different improvement, based on the observation that the decoding algorithm corrects an error vector based not so much on the weight of the vector but rather the ''footprint'' of the error locations. In both cases one can find minimal sets of parity checks defining the codes by means of the order function. In this paper we show that these minimal sets have a very useful closure property. For several important families of codes that we consider, this property allows us to construct a generating matrix for the code that has properties amenable to encoding. The generating matrix can be constructed by evaluating monomials in a set which also has the closure property.
2008, 2(1): 35-54
doi: 10.3934/amc.2008.2.35
+[Abstract](2292)
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Abstract:
We further the study of the duality theory of linear space-time codes over finite fields by showing that the only finite linear ''omnibus'' codes (defined herein) with a duality theory are the column distance codes and the rank codes. We introduce weight enumerators for both these codes and show that they have MacWilliams-type functional equations relating them to the weight enumerators of their duals. We also show that the complete weight enumerator for finite linear sum-of-ranks codes satisfies such a functional equation. We produce an analogue of Gleason's Theorem for formally self-dual linear finite rank codes. Finally, we relate the duality matrices of $n\times n$ linear finite rank codes and length $n$ vector codes under the Hamming metric.
We further the study of the duality theory of linear space-time codes over finite fields by showing that the only finite linear ''omnibus'' codes (defined herein) with a duality theory are the column distance codes and the rank codes. We introduce weight enumerators for both these codes and show that they have MacWilliams-type functional equations relating them to the weight enumerators of their duals. We also show that the complete weight enumerator for finite linear sum-of-ranks codes satisfies such a functional equation. We produce an analogue of Gleason's Theorem for formally self-dual linear finite rank codes. Finally, we relate the duality matrices of $n\times n$ linear finite rank codes and length $n$ vector codes under the Hamming metric.
2008, 2(1): 55-81
doi: 10.3934/amc.2008.2.55
+[Abstract](2069)
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Abstract:
In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the solvability of a modified rook problem. It is our strong belief that the rook problem is always solvable, and we present solutions in particular cases.
In this paper, we study convolutional codes with a specific cyclic structure. By definition, these codes are left ideals in a certain skew polynomial ring. Using that the skew polynomial ring is isomorphic to a matrix ring we can describe the algebraic parameters of the codes in a more accessible way. We show that the existence of such codes with given algebraic parameters can be reduced to the solvability of a modified rook problem. It is our strong belief that the rook problem is always solvable, and we present solutions in particular cases.
2008, 2(1): 83-94
doi: 10.3934/amc.2008.2.83
+[Abstract](2247)
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Abstract:
In this note we introduce the concept of group convolutional code in terms of skew polynomial rings. Then we use some theorems due to Jategaonkar that describe such rings in terms of matrices to characterize minimal $S_3$-convolutional codes over the field of five elements.
In this note we introduce the concept of group convolutional code in terms of skew polynomial rings. Then we use some theorems due to Jategaonkar that describe such rings in terms of matrices to characterize minimal $S_3$-convolutional codes over the field of five elements.
2008, 2(1): 95-111
doi: 10.3934/amc.2008.2.95
+[Abstract](2531)
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Abstract:
Let $P = (${$1, 2,\ldots,n$}$,$≤$)$ be a poset, let $V_1, V_2,\ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\mathbb F_q$ and let
$ V = V_1 \oplus V_2 \oplus\ldots \oplus V_n.$
In this paper we endow $V$ with a poset metric such that the $P$-weight is constant on the non-null vectors of a component $V_i$, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code $[8; 4; 4]$ to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code $[24; 12; 8]$ into perfect codes. We also give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometries of the error-block metric spaces.
Let $P = (${$1, 2,\ldots,n$}$,$≤$)$ be a poset, let $V_1, V_2,\ldots, V_n$ be a family of finite-dimensional spaces over a finite field $\mathbb F_q$ and let
$ V = V_1 \oplus V_2 \oplus\ldots \oplus V_n.$
In this paper we endow $V$ with a poset metric such that the $P$-weight is constant on the non-null vectors of a component $V_i$, extending both the poset metric introduced by Brualdi et al. and the metric for linear error-block codes introduced by Feng et al.. We classify all poset block structures which admit the extended binary Hamming code $[8; 4; 4]$ to be a one-perfect poset block code, and present poset block structures that turn other extended Hamming codes and the extended Golay code $[24; 12; 8]$ into perfect codes. We also give a complete description of the groups of linear isometries of these metric spaces in terms of a semi-direct product, which turns out to be similar to the case of poset metric spaces. In particular, we obtain the group of linear isometries of the error-block metric spaces.
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