New construction of M -ary sequence families with low correlation from the structure of Sidelnikov sequences

The main topics of this paper are the structure of Sidelnikov sequences and new construction of M ary sequence families from the structure. For prime p and a positive integer m, it is shown that M -ary Sidelnikov sequences of period p − 1, if M | pm − 1, can be equivalently generated by the operation of elements in a finite field GF(p), including a p-ary m-sequence. The equivalent representation over GF(pm) requires low complexity for implementing the Sidelnikov sequences of period p − 1. From the (pm − 1) × (pm + 1) array structure of the sequences, it is then found that a half of the column sequences and their constant multiples have low correlation enough to construct new M -ary sequence families of period pm − 1. In particular, new M -ary sequence families of period pm − 1 are constructed from the combination of the column sequence families and known Sidelnikov-based sequence families, where the new families have larger family sizes than the known ones with the same maximum correlation magnitudes. Finally, it is shown that the new M -ary sequence family of period pm−1 and the maximum correlation magnitude 2 √ p +6 asymptotically achieves √ 2 times the equality of the Sidelnikov’s lower bound when M = pm − 1 for odd prime p.