Transverse-read is a novel technique to detect the number of ‘1’s stored in a domain wall memory, also known as racetrack memory, without shifting any domains. Motivated by the technique, we propose a novel scheme to combine transverse-read and shift-operation such that we can reduce the number of shift-operations while still achieving high capacity. We also show that this scheme is helpful to correct errors in domain wall memory. A set of valid-words in this transverse-read channel is called a transverse-read code. Our goal in this work is to study transverse-read codes with respect to their properties, capacity, and applications. We first present several properties of transverse-read codes and show that they are equivalent to a family of constrained codes. Then, we compute the maximal asymptotic rate of transverse-read codes for certain parameters. Furthermore, we also present several constructions of transverse-read codes with high rate. Finally, we design several transverse-read codes that can correct limited-shift-errors and limited-magnitude errors in domain wall memory.