An information - theoretic proof of Hadamard's inequality

only linearly and not exponentially with the source block length. Since a trellis approaches a tree as the constraint length grows large, this work also suggests an alternate tree coding scheme and proof of the tree coding theorem of Jakatdar and Pearlman [6]. APPENDIX A The generalized Gallager function Edk(p) is defined in (21). In the following we prove that given RN,* > RN-*(De) for allj and k; or equivalently given (7a), that the per-letfer " rate " associated with each code letter being always greater than the rate r,(d,) induced by the rate-distortion function of the corresponding source letter u,, will imply (22a), that is Ep I 1-R P '0, foralljandkand-lO-1 < p < 0, for r > r,(dO) (28) where E,(p) and r,(do) are respectively the Gallager function and the rate-distortion function associated with the letter u,. We will use the property (28) to establish (27) as follows: