Bounds on the capacity of the bit-shift magnetic recording channel

The capacity region for W(write) U(unidirectiona1)Mbemory) codes over an alphabet of size q, for three of the cases asdefined by Simonyi are derived. For the fourth case (encoder anddecoder not informed about the state) an achievable rate region isderived that is believed to be the capacity region. Index Tenns -Capacity, write unidirectional memory. I. INTRODUCIIONWrite-unidirectional memory (WUM) codes have been pro-posed for use with rewritable optical disks. Recording of infor-mation on such a disk is based on the Kerr or Faraday effect [l]:Depending on the magnetization of a spot on the disk, apolarized laser beam directed at the spot suffers a rotation ofthe plane of polarization, which can be detected as0 or 1.Writing involves setting the direction of magnetization for eachspot and heating the spot with a laser. For this reason it wasproposed to fix the direction of magnetization for all spots onthe disk (e.g., to writing 0’s) and select the spots where to writeusing the laser. We call this a 0-cycle. When the next writing onthe disk takes place, the magnetization is reversed to writing 1’s(a 1-cycle). The 0and 1-cycles are executed alternatingly.Four cases have been studied (cf. [9]), depending on whetherencoder and/or decoder knows the state of the disk (or WUM)before writing. The capacity of the WUM is known for most ofthe cases (cf. [2],[3], [6], [lo], [ll]).In this correspondence wegeneralize theWUM to an alphabet d with /dl= q. Here wehave q types of cycles, executed cyclically, where in a c-cycle( c E d )only c’s can be written. We show that the results forthe capacity of the binary WUM can be generalized to the q-arycase.Although we study the q-ary WUM mainly because of itsmathematical interest, it is conceivable that the model is of usefor multileveled recording systems in the future. After all, thephysical processes that play a role in the storage of informationin material are seldom of a binary nature. In fact, these pro-cesses are almost always continuous: the depth of a pit inwrite-once media, the color change in certain polymers, and alsomagnetization (both in absolute value and in direction).As an example, the rotation of the plane of polarizationpreviously described is proportional to the magnetic field [ 11. Manuscript received October 13, 1989; revised July 4, 1990. This workwas presented in part at the International Symposium on InformationTheory, San Diego, CA, January 14-19, 1990.The author is with the Institute for Perception Research, IPO, P.O.Box 513, 5600MB Eindhoven, The Netherlands.IEEE Log Number 9142950. Oo18-9448/91/05~-0872$01.OO 01991 IEEE