The result presented here will speed up certain elementary calculations. The author gives a clear exposition of the problem and proof. The paper is relatively easy reading for those with a sound understanding and the necessary mathematical background.
Hadamard matrices are m × m matrices with all elements ∓ 1 and with the property that H T H = m &dundot; I. Two such matrices are equivalent if row and column changes convert one into the other. A special equivalence class when m = 2 n is the Walsh class, defined by its containing the matrix W n, where
A binary m-sequence is a sequence of bits a i in which for all k. For a suitable choice of the c i, this sequence has period 2 n - 1, provided that not all the initial a i are zero. Binary m-sequences have been used extensively as pseudorandom sequences for appropriate applications. By noting that the set of distinct sequences of n bits that can be made from the nonidentity part of an Abelian group is isomorphic to ( Z 2 ) n and that Walsh matrices have a similar construction (albeit multiplicative instead of additive), the problem of computing some binary m-sequences reduces to writing down Walsh matrices. Thus the computation of m-sequences can be reduced to a single Fast Walsh Transform, cutting computation times substantially.
The proof of the basic result is neat but elementary and not entirely surprising. It is, therefore, surprising that the work has not been done before. The paper contains a few misprints, but careful reading will enable the reader to put them right. The reference list is appropriate and good.
