PYRAMID SCHEME FOR CONSTRUCTING BIORTHOGONAL WAVELET CODES OVER FINITE FIELDS

Abstract

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⊕. . .⊕WJ⊕VJ and V = W˜0⊕W˜1⊕. . .⊕W˜J⊕V˜J ,such that at the j-th level of the decomposition, for 0 < j 6 J, Vj−1 = Vj⊕Wj , V˜j−1 == V˜j ⊕ W˜j , the subspace Vj 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 (hj, gj) 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.