4-phase Sequences with Near-optimum Correlation Properties

Two families of 4-phase sequences are constructed using irreducible polynomials over Z4. Family d has period L = 2' 1, size L +2, and maximum nontrivial correlation magnitude C,,, 5 1 + m, where r is a positive integer. Family has period L = 2(2' l), size ( L + 2)/4, and C,,, 5 2 + m. Both families are asymptotically optimal with respect to the Welch lower bound on C,,, for complex-valued sequences. Of particular interest, Family d has the same size and period as the family of binary Gold sequences, but its maximum nontrivial correlation is smaller by a factor of 4. Since the Gold family for r odd is optimal with respect to the Welch bound restricted to binary sequences, Family d is thus superior to the best possible binary design of the same family size. Unlike the Gold design, Families d and ! t ? are asymptotically optimal whether r is odd or even. Both families are suitable for achieving code-division multiple-access and are easily implemented using shift registers. The exact distribution of correlation values is given for both families.