Perfect Codes and Balanced Generalized Weighing Matrices, II

In a previous paper, the authors proved that any set of representatives of the distinct 1-dimensional subspaces in the dual code of the unique linear perfect single-error-correcting code of length (qd−1)/(q−1) over GF(q) is a balanced generalized weighing matrix over the multiplicative group of GF(q). Moreover, this matrix was characterized as the unique (up to monomial equivalence) weighing matrix for the given parameters with minimum q-rank. We now relate these matrices to m-sequences (that is, linear shift register sequences of maximal period) by giving an explicit description in terms of the trace function; in this way, we show that a simple modification of our method can be used to obtain the matrices which are given by the “classical,” more involved construction going back to Berman. Moreover, further modifications of our matrices actually yield a wealth of monomially inequivalent examples, namely matrices for many different q-ranks.