Pyramid scheme for constructing biorthogonal wavelet codes over finite fields

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.