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 Prikl. Diskr. Mat.: Year: Volume: Issue: Page: Find

 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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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}