ISSN:
1930-5346
eISSN:
1930-5338
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Advances in Mathematics of Communications
August 2012 , Volume 6 , Issue 3
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2012, 6(3): 259-272
doi: 10.3934/amc.2012.6.259
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Abstract:
A list decoding algorithm for matrix-product codes is provided when $C_1, ..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.
A list decoding algorithm for matrix-product codes is provided when $C_1, ..., C_s$ are nested linear codes and $A$ is a non-singular by columns matrix. We estimate the probability of getting more than one codeword as output when the constituent codes are Reed-Solomon codes. We extend this list decoding algorithm for matrix-product codes with polynomial units, which are quasi-cyclic codes. Furthermore, it allows us to consider unique decoding for matrix-product codes with polynomial units.
2012, 6(3): 273-285
doi: 10.3934/amc.2012.6.273
+[Abstract](4041)
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Abstract:
In 1998 Crandall introduced a method based on coding theory to secretly embed a message in a digital support such as an image. Later, in 2005, Fridrich et al. improved this method to minimize the distortion introduced by the embedding; a process called wet paper. However, as previously emphasized in the literature, this method can fail during the embedding step. Here we find sufficient and necessary conditions to guarantee a successful embedding, by studying the dual distance of a linear code. Since these results are essentially of combinatorial nature, they can be generalized to systematic codes, a large family containing all linear codes. We also compute the exact number of embedding solutions and point out the relationship between wet paper codes and orthogonal arrays.
In 1998 Crandall introduced a method based on coding theory to secretly embed a message in a digital support such as an image. Later, in 2005, Fridrich et al. improved this method to minimize the distortion introduced by the embedding; a process called wet paper. However, as previously emphasized in the literature, this method can fail during the embedding step. Here we find sufficient and necessary conditions to guarantee a successful embedding, by studying the dual distance of a linear code. Since these results are essentially of combinatorial nature, they can be generalized to systematic codes, a large family containing all linear codes. We also compute the exact number of embedding solutions and point out the relationship between wet paper codes and orthogonal arrays.
2012, 6(3): 287-303
doi: 10.3934/amc.2012.6.287
+[Abstract](3923)
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Abstract:
Self-dual codes over $\mathbb Z_2\times\mathbb Z_4$ are subgroups of $\mathbb Z_2^\alpha\times\mathbb Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates these codes to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exist a self-dual code $\mathcal C\subseteq \mathbb Z_2^\alpha \times\mathbb Z_4^\beta$ are established. Moreover, the construction of such a code for each type and possible pair $(\alpha,\beta)$ is given. The standard techniques of invariant theory are applied to describe the weight enumerators for each type. Finally, we give a construction of self-dual codes from existing self-dual codes.
Self-dual codes over $\mathbb Z_2\times\mathbb Z_4$ are subgroups of $\mathbb Z_2^\alpha\times\mathbb Z_4^\beta$ that are equal to their orthogonal under an inner-product that relates these codes to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $\alpha,\beta$ such that there exist a self-dual code $\mathcal C\subseteq \mathbb Z_2^\alpha \times\mathbb Z_4^\beta$ are established. Moreover, the construction of such a code for each type and possible pair $(\alpha,\beta)$ is given. The standard techniques of invariant theory are applied to describe the weight enumerators for each type. Finally, we give a construction of self-dual codes from existing self-dual codes.
2012, 6(3): 305-314
doi: 10.3934/amc.2012.6.305
+[Abstract](3323)
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Abstract:
Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to construct Maiorana-McFarland's (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.
Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to construct Maiorana-McFarland's (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.
2012, 6(3): 315-328
doi: 10.3934/amc.2012.6.315
+[Abstract](3072)
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Abstract:
In this work we present a canonical-systematic form of a generator matrix for linear codes whith respect to a hierarchical poset metric on the linear space $\mathbb F_q^n$. We show that up to a linear isometry any such code is equivalent to the direct sum of codes with smaller dimensions. The canonical-systematic form enables to exhibit simple expressions for the generalized minimal weights (in the sense defined by Wei), the packing radius of the code, characterization of perfect codes and also syndrome decoding algorithm that has (in general) exponential gain when compared to usual syndrome decoding.
In this work we present a canonical-systematic form of a generator matrix for linear codes whith respect to a hierarchical poset metric on the linear space $\mathbb F_q^n$. We show that up to a linear isometry any such code is equivalent to the direct sum of codes with smaller dimensions. The canonical-systematic form enables to exhibit simple expressions for the generalized minimal weights (in the sense defined by Wei), the packing radius of the code, characterization of perfect codes and also syndrome decoding algorithm that has (in general) exponential gain when compared to usual syndrome decoding.
2012, 6(3): 329-346
doi: 10.3934/amc.2012.6.329
+[Abstract](3511)
+[PDF](467.9KB)
Abstract:
The paper deals with some structural properties of propelinear binary codes, in particular propelinear perfect binary codes. We consider the connection of transitive codes with propelinear codes and show that there exists a binary code, the Best code of length 10, size 40 and minimum distance 4, which is transitive but not propelinear. We propose several constructions of propelinear codes and introduce a new large class of propelinear perfect binary codes, called normalized propelinear perfect codes. Finally, based on the different values for the rank and the dimension of the kernel, we give a lower bound on the number of nonequivalent propelinear perfect binary codes.
The paper deals with some structural properties of propelinear binary codes, in particular propelinear perfect binary codes. We consider the connection of transitive codes with propelinear codes and show that there exists a binary code, the Best code of length 10, size 40 and minimum distance 4, which is transitive but not propelinear. We propose several constructions of propelinear codes and introduce a new large class of propelinear perfect binary codes, called normalized propelinear perfect codes. Finally, based on the different values for the rank and the dimension of the kernel, we give a lower bound on the number of nonequivalent propelinear perfect binary codes.
2012, 6(3): 347-361
doi: 10.3934/amc.2012.6.347
+[Abstract](3117)
+[PDF](373.8KB)
Abstract:
In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb F_q$. For the first time Möbius and Rédei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of Rédei functions an exact formula for the inverse function is derived. The cycle structure of Rédei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers.
In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb F_q$. For the first time Möbius and Rédei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of Rédei functions an exact formula for the inverse function is derived. The cycle structure of Rédei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers.
2012, 6(3): 363-384
doi: 10.3934/amc.2012.6.363
+[Abstract](2444)
+[PDF](438.2KB)
Abstract:
This paper considers the problem of cross-moments computation for functions which decompose according to cycle-free factor graphs. Two algorithms are derived, both based on message passing computation of a corresponding moment-generating function ($MGF$). The first one is realized as message passing algorithm over a polynomial semiring and represents a computation of the $MGF$ Taylor coefficients, while the second one represents message passing algorithm over a binomial semiring and a computation of the $MGF$ partial derivatives. We found that some previously developed algorithms can be seen as special cases of our algorithms and we consider the time and memory complexities.
This paper considers the problem of cross-moments computation for functions which decompose according to cycle-free factor graphs. Two algorithms are derived, both based on message passing computation of a corresponding moment-generating function ($MGF$). The first one is realized as message passing algorithm over a polynomial semiring and represents a computation of the $MGF$ Taylor coefficients, while the second one represents message passing algorithm over a binomial semiring and a computation of the $MGF$ partial derivatives. We found that some previously developed algorithms can be seen as special cases of our algorithms and we consider the time and memory complexities.
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