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For the case where p is an odd prime, n>or=2 is an integer, and omega is a complex primitive pth root of unity, a construction is presented for a family of pn p-phase sequences (symbols of the form omega i), where each sequence has length pn-1, and where the maximum nontrivial correlation value Cmax does not exceed 1+ square root pn. A complete distribution of correlation values is provided. As a special case of this construction, a previous construction due to Sidelnikov (1971) is obtained. The family of sequences is asymptotically optimum with respect to its correlation properties, and, in comparison with many previous nonbinary designs, the present design has the additional advantage of not requiring an alphabet of size larger than three. The new sequences are suitable for achieving code-division multiple access and are easily implemented using shift registers. They wee discovered through an application of Deligne's bound (1974) on exponential sums of the Weil type in, several variables. The sequences are also shown to have strong identification with certain bent functions.