An image forming apparatus includes a latent image bearing member, a charging unit, a charging power source, a latent image writing unit, a development unit, a transfer unit, a toner detector, and a controller. An input to the S-box is converted from a Galois field representation GF to a Galois subfield representation GF2. , in a product sales data processing system can be saved. If it is desired to transfer one of the valid data symbols that has been mapped to an invalid symbol, the data symbol may be replaced with an unused symbol. Each nasal cannula includes an open recessed channel disposed between a pair of bridges for retaining the extension tube within the earpiece and an locking exit hole at its distal end for helping to secure therein the extension tube to provide a fixed distance adjustment between the nasal cannula and the ears of the patient. The method may also include evaluating an accuracy of the pre-fetch cache requests.