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Prikl. Diskr. Mat., 2021, Number 51, Pages 101–119 (Mi pdm733)  

Applied Coding Theory

Pyramid scheme for constructing biorthogonal wavelet codes over finite fields

D. V. Litichevskiy

Chelyabinsk state university, Chelyabinsk, Russia

Abstract: The existence of a biorthogonal decomposition of the space $V$ of dimension $n$ over the field $\mathrm{GF}(q)$ is constructively proved, namely, two representations of it are obtained as direct sums of subspaces $V =W_0 \oplus W_1 \oplus \ldots \oplus W_J \oplus V_J$ and $V = \tilde{W}_0 \oplus \tilde{W}_1 \oplus \ldots \oplus \tilde {W}_J \oplus \tilde{V}_J $, such that at the $j$-th level of the decomposition, for $0< j\leq J$, $V_{j-1}=V_j\oplus W_j$, $\tilde{V}_{j-1}= \tilde{V}_j\oplus \tilde{W}_j$, the subspace $V_j$ is orthogonal to $\tilde{W}_j $, and the subspace $W_j$ is orthogonal to $\tilde{V}_j $. The partition of the space at the $j$-th level is made with the help of pairs of level filters $(h^j, g^j)$ and $ (\tilde{h}^ j, \tilde{g}^j)$, for the construction of which the corresponding algorithms have been developed and theoretically proved. A new family of biorthogonal wavelet codes is built on the basis of the multilevel wavelet decomposition scheme with coding rate $2^{-L}$, where $L$ is the number of used decomposition levels, and examples of such codes are given.

Keywords: discrete biorthogonal wavelet transforms, multiresolutions, wavelet codes.

DOI: https://doi.org/10.17223/20710410/51/5

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Bibliographic databases:

UDC: 519.725

Citation: D. V. Litichevskiy, “Pyramid scheme for constructing biorthogonal wavelet codes over finite fields”, Prikl. Diskr. Mat., 2021, no. 51, 101–119

Citation in format AMSBIB
\Bibitem{Lit21}
\by D.~V.~Litichevskiy
\paper Pyramid scheme for constructing biorthogonal wavelet codes over finite fields
\jour Prikl. Diskr. Mat.
\yr 2021
\issue 51
\pages 101--119
\mathnet{http://mi.mathnet.ru/pdm733}
\crossref{https://doi.org/10.17223/20710410/51/5}


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