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Applicable Algebra in Engineering, Communication and Computing

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Erschienen in: Applicable Algebra in Engineering, Communication and Computing 1/2020

26.06.2019 | Original Paper

Properties of syndrome distribution for blind reconstruction of cyclic codes

verfasst von: Arti D. Yardi, Saravanan Vijayakumaran

Erschienen in: Applicable Algebra in Engineering, Communication and Computing | Ausgabe 1/2020

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Abstract

In the problem of blind reconstruction of channel codes, the receiver does not have the knowledge of the channel code used at the transmitter and the aim is to identify this unknown code from the received data. For the case of cyclic codes, typical blind reconstruction methods make use of some elementary properties of syndromes (remainders) of the received polynomials. The main aim of this paper is to provide a detailed analysis of the properties of the syndromes that could be useful to design more efficient blind reconstruction methods. Specifically, we prove that the syndrome distribution of the noise-free sequence can be either degenerate or uniform or restricted uniform. We also provide necessary and sufficient conditions for the syndrome distribution to be of a given type. For the noise-affected received sequence we identify additional structural properties exhibited by the syndrome distribution. Finally, we apply these results to analyze the performance of the existing blind reconstruction methods.
Vorheriger Artikel Arf good semigroups with fixed genus
Nächster Artikel Multidimensional linear complexity analysis of periodic arrays

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Fußnoten
1
In the literature, a potential low-weight codeword in the dual space can be chosen arbitrarily or can be obtained using information-set decoding algorithms [5, 9, 10].
 
2
A linear block code C(n) is said to be degenerate if its generator matrix G can be written as a concatenation of two or more identical copies of a generator matrix \(G^{\prime }\) of some other linear block code [20, Ch. 8].
 
3
\(C_1(n_1) + C_2(n_2) :=\Big \{ [{\mathbf {c}}_1 {~} {\mathbf {c}}_2] \text{ such } \text{ that } {\mathbf {c}}_1 \in C_1(n_1) \text{ and } {\mathbf {c}}_2 \in C_2(n_2) \Big \}\) [15].
 
4
Assumed length n is correct up to an integer multiple of \(n_0\).
 
5
The dual of the code obtained by puncturing C(n) at any d coordinate locations is equal to the code obtained by shortening \(C^{\perp }(n)\) at the same locations [23, Section 3.2.5].
 
6
When \(p=0\), \({\mathbb {P}}[{\mathbf {e}}_j(X) \in C(n,f)] = 1\) and \({\mathbb {P}}[{\mathbf {e}}_j(X) \in {\mathcal {G}}_d(n,f)] = 0\) for any \({\mathcal {G}}_d(n,f)\).
 
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Metadaten
Titel
Properties of syndrome distribution for blind reconstruction of cyclic codes
verfasst von
Arti D. Yardi
Saravanan Vijayakumaran
Publikationsdatum
26.06.2019
Verlag
Springer Berlin Heidelberg
Erschienen in
Applicable Algebra in Engineering, Communication and Computing / Ausgabe 1/2020
Print ISSN: 0938-1279
Elektronische ISSN: 1432-0622
DOI
https://doi.org/10.1007/s00200-019-00392-0

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