1. An information recording and reproducing apparatus comprising: recording means having first selection means for selecting user data and training data based on a recording address, modulation means for creating a plurality of modulation data candidates with respect to data selected by this first selection means, second selection means for selecting various modulation data candidates created by this modulation means, a counter for counting DSVs in modulation data selected by this second selection means, and control means for setting a DSV output from this counter to a specified value with said training data selected by said first selection means, wherein said recording means records modulation data selected by said second selection means on an information recording medium; and reproduction means having equalization means for equalizing a reproduction signal read from said information recording medium based on an equalization coefficient, discrimination means for discriminating output from this equalization means, training data generation means for outputting predetermined training data based on output from this discrimination means, polarity test means for testing a polarity of training data output from this training data generation means based on output discrimination data of said discrimination means, ideal signal generation means for generating an ideal signal when the training data is ideally equalized from output of this polarity test means, operation means for computing an equalization error based on output from this ideal signal generation means and output from said equalization means, and coefficient control means for generating an equalization coefficient supplied to said equalization means based on output from this operation means and a reproduction signal input to said equalization means.