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1930-5346
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Advances in Mathematics of Communications
May 2012 , Volume 6 , Issue 2
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2012, 6(2): 121-130
doi: 10.3934/amc.2012.6.121
+[Abstract](2144)
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Abstract:
It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank $r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length $n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.
It is proved that for every integer $n=2^k-1$, with $k\geq5$, there exists a perfect code $C$ of length $n$, of rank $r=n-\log(n+1)+2$ and with a trivial symmetry group. This result extends an earlier result by the authors that says that for any length $n=2^k-1$, with $k\geq5$, and any rank $r$, with $n-\log(n+1)+3\leq r\leq n-1$ there exist perfect codes with a trivial symmetry group.
2012, 6(2): 131-163
doi: 10.3934/amc.2012.6.131
+[Abstract](2224)
+[PDF](620.8KB)
Abstract:
Codes defined on graphs and their properties have been subjects of intense recent research. In this work, we are concerned with codes that have planar Tanner graphs. When the Tanner graph is planar, message-passing decoders can be efficiently implemented on chips without any issues of wiring. Also, recent work has shown the existence of optimal decoders for certain planar graphical models. The main contribution of this paper is an explicit upper bound on minimum distance $d$ of codes that have planar Tanner graphs as a function of the code rate $R$ for $R \geq 5/8$. The bound is given by \begin{equation*} d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3\le 7. \end{equation*} As a result, high-rate codes with planar Tanner graphs will result in poor block-error rate performance, because of the constant upper bound on minimum distance.
Codes defined on graphs and their properties have been subjects of intense recent research. In this work, we are concerned with codes that have planar Tanner graphs. When the Tanner graph is planar, message-passing decoders can be efficiently implemented on chips without any issues of wiring. Also, recent work has shown the existence of optimal decoders for certain planar graphical models. The main contribution of this paper is an explicit upper bound on minimum distance $d$ of codes that have planar Tanner graphs as a function of the code rate $R$ for $R \geq 5/8$. The bound is given by \begin{equation*} d\le \left\lceil \frac{7-8R}{2(2R-1)} \right\rceil + 3\le 7. \end{equation*} As a result, high-rate codes with planar Tanner graphs will result in poor block-error rate performance, because of the constant upper bound on minimum distance.
2012, 6(2): 165-174
doi: 10.3934/amc.2012.6.165
+[Abstract](2458)
+[PDF](345.4KB)
Abstract:
Batch codes, introduced by Ishai et al. in [11], are methods for solving the following data storage problem: $n$ data items are to be stored in $m$ servers in such a way that any $k$ of the $n$ items can be retrieved by reading at most $t$ items from each server, and that the total number of items stored in $m$ servers is $N$. A combinatorial batch code (CBC) is a batch code where each data item is stored without change, i.e., each stored data item is a copy of one of the $n$ data items.
One of the basic yet challenging problems is to find optimal CBCs, i.e., CBCs for which total storage ($N$) is minimal for given values of $n$, $m$, $k$, and $t$. In [13], Paterson et al. exclusively studied CBCs and gave constructions of some optimal CBCs.
In this article, we give a lower bound on the total storage ($N$) for CBCs. We give explicit construction of optimal CBCs for a range of values of $n$. For a different range of values of $n$, we give explicit construction of optimal and almost optimal CBCs. Our results partly settle an open problem of [13].
Batch codes, introduced by Ishai et al. in [11], are methods for solving the following data storage problem: $n$ data items are to be stored in $m$ servers in such a way that any $k$ of the $n$ items can be retrieved by reading at most $t$ items from each server, and that the total number of items stored in $m$ servers is $N$. A combinatorial batch code (CBC) is a batch code where each data item is stored without change, i.e., each stored data item is a copy of one of the $n$ data items.
One of the basic yet challenging problems is to find optimal CBCs, i.e., CBCs for which total storage ($N$) is minimal for given values of $n$, $m$, $k$, and $t$. In [13], Paterson et al. exclusively studied CBCs and gave constructions of some optimal CBCs.
In this article, we give a lower bound on the total storage ($N$) for CBCs. We give explicit construction of optimal CBCs for a range of values of $n$. For a different range of values of $n$, we give explicit construction of optimal and almost optimal CBCs. Our results partly settle an open problem of [13].
2012, 6(2): 175-191
doi: 10.3934/amc.2012.6.175
+[Abstract](2490)
+[PDF](408.6KB)
Abstract:
The polynomial residue ring $\mathcal R_a=\frac{\mathbb F_{2^m}[u]}{\langle u^a \rangle}=\mathbb F_{2^m} + u \mathbb F_{2^m}+ \dots + u^{a - 1}\mathbb F_{2^m}$ is a chain ring with residue field $\mathbb F_{2^m}$, that contains precisely $(2^m-1)2^{m(a-1)}$ units, namely, $\alpha_0+u\alpha_1+\dots+u^{a-1}\alpha_{a-1}$, where $\alpha_0,\alpha_1,\dots,\alpha_{a-1} \in \mathbb F_{2^m}$, $\alpha_0 \neq 0$. Two classes of units of $\mathcal R_a$ are considered, namely, $\lambda=1+u\lambda_1+\dots+u^{a-1}\lambda_{a-1}$, where $\lambda_1, \dots, \lambda_{a-1} \in \mathbb F_{2^m}$, $\lambda_1 \neq 0$; and $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{2^m}$, $\Lambda_0 \neq 0, \Lambda_1 \neq 0$. Among other results, the structure, Hamming and homogeneous distances of $\Lambda$-constacyclic codes of length $2^s$ over $\mathcal R_a$, and the structure of $\lambda$-constacyclic codes of any length over $\mathcal R_a$ are established.
The polynomial residue ring $\mathcal R_a=\frac{\mathbb F_{2^m}[u]}{\langle u^a \rangle}=\mathbb F_{2^m} + u \mathbb F_{2^m}+ \dots + u^{a - 1}\mathbb F_{2^m}$ is a chain ring with residue field $\mathbb F_{2^m}$, that contains precisely $(2^m-1)2^{m(a-1)}$ units, namely, $\alpha_0+u\alpha_1+\dots+u^{a-1}\alpha_{a-1}$, where $\alpha_0,\alpha_1,\dots,\alpha_{a-1} \in \mathbb F_{2^m}$, $\alpha_0 \neq 0$. Two classes of units of $\mathcal R_a$ are considered, namely, $\lambda=1+u\lambda_1+\dots+u^{a-1}\lambda_{a-1}$, where $\lambda_1, \dots, \lambda_{a-1} \in \mathbb F_{2^m}$, $\lambda_1 \neq 0$; and $\Lambda=\Lambda_0+u\Lambda_1+\dots+u^{a-1}\Lambda_{a-1}$, where $\Lambda_0, \Lambda_1, \dots, \Lambda_{a-1} \in \mathbb F_{2^m}$, $\Lambda_0 \neq 0, \Lambda_1 \neq 0$. Among other results, the structure, Hamming and homogeneous distances of $\Lambda$-constacyclic codes of length $2^s$ over $\mathcal R_a$, and the structure of $\lambda$-constacyclic codes of any length over $\mathcal R_a$ are established.
2012, 6(2): 193-202
doi: 10.3934/amc.2012.6.193
+[Abstract](2283)
+[PDF](304.8KB)
Abstract:
In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructions for self-dual codes over the non-chain ring $R_2 = \mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$ are introduced. Using these methods, we have constructed three extremal binary Type I codes of length $64$ of new weight enumerators for which extremal codes were not known to exist. We also give a double-circulant construction for extremal binary self-dual codes of length $40$ with covering radius $7$.
In this work, the double-circulant, bordered-double-circulant and stripped bordered-double-circulant constructions for self-dual codes over the non-chain ring $R_2 = \mathbb F_2+u\mathbb F_2+v\mathbb F_2+uv\mathbb F_2$ are introduced. Using these methods, we have constructed three extremal binary Type I codes of length $64$ of new weight enumerators for which extremal codes were not known to exist. We also give a double-circulant construction for extremal binary self-dual codes of length $40$ with covering radius $7$.
2012, 6(2): 203-227
doi: 10.3934/amc.2012.6.203
+[Abstract](2324)
+[PDF](596.2KB)
Abstract:
The random matrix uniformly distributed over the set of all $m$-by-$n$ matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called $k$-good random matrices, is studied. It is shown that a $k$-good random $m$-by-$n$ matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{$m,n$}$-k+1$, and vice versa. Further examples of $k$-good random matrices are derived from homogeneous weights on matrix modules. Several applications of $k$-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a $k$-dense set of $m$-by-$n$ matrices is studied, identifying such sets as blocking sets with respect to $(m-k)$-dimensional flats in a certain $m$-by-$n$ matrix geometry and determining their minimum size in special cases.
The random matrix uniformly distributed over the set of all $m$-by-$n$ matrices over a finite field plays an important role in many branches of information theory. In this paper a generalization of this random matrix, called $k$-good random matrices, is studied. It is shown that a $k$-good random $m$-by-$n$ matrix with a distribution of minimum support size is uniformly distributed over a maximum-rank-distance (MRD) code of minimum rank distance min{$m,n$}$-k+1$, and vice versa. Further examples of $k$-good random matrices are derived from homogeneous weights on matrix modules. Several applications of $k$-good random matrices are given, establishing links with some well-known combinatorial problems. Finally, the related combinatorial concept of a $k$-dense set of $m$-by-$n$ matrices is studied, identifying such sets as blocking sets with respect to $(m-k)$-dimensional flats in a certain $m$-by-$n$ matrix geometry and determining their minimum size in special cases.
2012, 6(2): 229-235
doi: 10.3934/amc.2012.6.229
+[Abstract](2967)
+[PDF](306.4KB)
Abstract:
A complete classification of binary self-dual codes of length $36$ is given.
A complete classification of binary self-dual codes of length $36$ is given.
2012, 6(2): 237-247
doi: 10.3934/amc.2012.6.237
+[Abstract](2700)
+[PDF](483.5KB)
Abstract:
A family of quaternary periodic complementary sequence (PCS) or Z-complementary sequence (PZCS) sets is presented. By combining an interleaving technique and the inverse Gray mapping, the proposed method transforms the known binary PCS/PZCS sets with odd length of sub-sequences into quaternary PCS/PZCS sets, but the length of new sub-sequences is twice as long as that of the original sub-sequences, which is a drawback of this proposed method. However, the shortcoming that the method proposed by J. W. Jang, et al. is merely fit for even length of sub-sequences is overcome. As a consequence, the union of our and J. W. Jang, et al.'s methods allows us to construct quaternary PCS/PZCS sets from binary PCS/PZCS sets with sub-sequences of arbitrary length.
A family of quaternary periodic complementary sequence (PCS) or Z-complementary sequence (PZCS) sets is presented. By combining an interleaving technique and the inverse Gray mapping, the proposed method transforms the known binary PCS/PZCS sets with odd length of sub-sequences into quaternary PCS/PZCS sets, but the length of new sub-sequences is twice as long as that of the original sub-sequences, which is a drawback of this proposed method. However, the shortcoming that the method proposed by J. W. Jang, et al. is merely fit for even length of sub-sequences is overcome. As a consequence, the union of our and J. W. Jang, et al.'s methods allows us to construct quaternary PCS/PZCS sets from binary PCS/PZCS sets with sub-sequences of arbitrary length.
2012, 6(2): 249-258
doi: 10.3934/amc.2012.6.249
+[Abstract](2420)
+[PDF](330.3KB)
Abstract:
A class of bent functions on a Galois ring is introduced and based on these functions systematic authentication codes are presented. These codes generalize those appearing in [4] for finite fields.
A class of bent functions on a Galois ring is introduced and based on these functions systematic authentication codes are presented. These codes generalize those appearing in [4] for finite fields.
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