A proof of non-existence of bordered jacket matrices of odd order over some fields

Lee and colleagues have introduced a generalised reverse jacket transform (GRJT) as a multi-phase or multilevel generalisation of the WHT and the even-length DFT. The matrix representing a primary GRJT is permutation-equivalent to the DFT matrix of even order, and in addition possesses the property to have a border consisting entirely of ??1´s. To the authors ´ knowledge, the problem of existence of such transform matrices in the case of odd order, has attracted minor attention in the existing literature. In this Letter, a contribution is made regarding this problem in respect to an extended class of matrices.