Pyramid scheme for constructing biorthogonal wavelet codes over finite fields | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2021. № 51. DOI: 10.17223/20710410/51/5

The existence of a biorthogonal decomposition of the space V of dimension n over the field GF(q) is constructively proved, namely, two representations of it are obtained as direct sums of subspaces V = W₀⊕W₁⊕.. .⊕W_J⊕V_J и V = Wо ⊕ Wi ⊕ ... ⊕ W_J ⊕ V_J, such that at the j-th level of the decomposition, for (0 < j ≤ J), V_j-1 = Vj ⊕ W_j, V_j-1 = V_j ⊕ W_j, the subspace V_j is orthogonal to W_j, and the subspace Wj is orthogonal to 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 (h^j,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.