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This paper presents the optimum random access channel (RACH) structure in terms of the number of control signaling bits accommodated and the transmission bandwidth based on the link budget in order to satisfy the coverage requirement for the single-carrier (SC)-FDMA based E-UTRA uplink. First, we propose grouping of the RACH purposes to reduce the number of control signaling bits to be conveyed by the RACH. By employing the proposed purpose-grouping scheme, only a four-bit random ID and a two-bit purpose grouping index for the non-synchronized RACH are required. Simulation evaluations including a system-level simulation show that the non-synchronized RACH comprising a 0.5-msec preamble signature sequence can convey the aforementioned 6-bit control signal at the cell edge when the inter-site distance (ISD) is 500 m under full channel load conditions with one-cell frequency reuse. It is also shown, however, that a RACH longer than one-sub-frame, e.g., 1.0 msec, is necessary to support the ISD of 1732 m assuming the same conditions. We also show that although there is no distinct difference in the transmission bandwidth from 1.25 to 5 MHz from the viewpoint of the misdetection probability, a 1.25-MHz transmission bandwidth for the RACH is suitable from the viewpoints of flexible resource assignment in the time domain and small number of option in the transmission bandwidth.