On the histrograms of the correlation functions of product codes

This paper studies the correlation properties of a family of product codes, obtained as the product of two distinct m-sequences. First we define a set of primary product codes in terms of the two m-sequences. The collection of the auto-correlation functions of the primary product codes are divided into ensembles. It is shown that these ensembles are cyclic shifted versions of the cross-correlation function between the two m-sequences that generate the product code. The relationships between these ensembles are also derived. The auto-correlation histograms of a set of primary product codes corresponding to an integer class are shown to be equal. For the special case of product codes referred to as the Gold codes, obtained as the product of two preferred m-sequences, the amount of unbalance in the autocorrelation histograms of the primary Gold codes are related to the cross-correlation between the preferred m-sequences.