In this paper, we show that the cross-correlation of two properly chosen column sequences from the array structure of two different Sidelnikov sequences of periods qe-1 and qf- 1, where e ≠ f, is bounded by (e+f-1)√q+1. From this result, we construct new sequence families by combining sequence families from the array structure of Sidelnikov sequences of period q2- 1, q3- 1,..., qd- 1 for some d with 2 ≤ d ≤ 1/2(√q - 2/√q + 1). The maximum non-trivial complex correlation of any two pair of sequences in the constructed sequence family is upper-bounded by (2d - 1)√q + 1: thus, the combining process does not affect the maximum non-trivial complex correlation.

We show that a p-ary polyphase sequence of period p2from the Fermat quotients is perfect. That is, its periodic autocorrelation is zero for all non-trivial phase shifts. We call this Fermat-quotient sequence. We propose a collection of optimal families of perfect polyphase sequences using the Fermatquotient sequences in the sense of the Sarwate bound. That is, the cross correlation of two members in a family is upper bounded by p. To investigate some relation between Fermat-quotient sequences and Frank-Zadoff sequences and to construct optimal families including these sequences, we introduce generators of p-ary polyphase sequences of period p2using their p × p array structures. We call an optimal generator to be the generator of some p-ary polyphase sequences which are perfect and which gives an optimal family by the proposed construction. Finally, we propose an algebraic construction for optimal generators as another main result. A lot of optimal families of size p - 1 can be constructed from these optimal generators, some of which are known to be from the Fermat-quotient sequences or from the Frank-Zadoff sequences, but some families are new for p ≥ 11. The relation between the Fermat-quotient sequences and the Frank-Zadoff sequences is determined as a by-product.

We show that a p-ary polyphase sequence of period p2from the Fermat quotients is `perfect.' That is, its periodic autocorrelation is zero for all non-trivial shifts. We call this Fermat-Quotient sequences. Using this and the fact that the Frank-Zadoff sequences (which is known to be also perfect), we propose a collection of `optimum' families of perfect polyphase sequences in the sense of Sarwate Bound. That is, the cross-correlation of two members in a family is upper bounded by p. We may say these families are `completely optimum' since the cross-correlation of any two members in a family is exactly p for all phase-shifts.

In this paper, we extend the construction by Vu and Gong for families of M-ary sequences of period q - 1 from the array structure of an M-ary Sidelnikov sequence of period q2- 1, where q is a prime power and M|q - 1. The construction now applies to the cases of using any period qd-1 for 3 ≤ d27. The proposed construction results in a family of M-ary sequences of period q-1 with: 1) the correlation magnitudes, which are upper bounded by (2d -1)√q +1 and 2) the asymptotic size of (M -1)qd-1/d as q increases. We also characterize some subsets of the above of size ~(r - 1)qd-1/d but with a tighter upper bound (2d - 2)√q + 2 on its correlation magnitude. We discuss reducing both time and memory complexities for the practical implementation of such constructions in some special cases. We further give some approximate size of the newly constructed families in general and an exact count when d is a prime power or a product of two distinct primes. The main results of this paper now give more freedom of tradeoff in the design of M-ary sequence family between the family size and the correlation magnitude of the family.

Let$p$be a prime and$n$a positive integer. Let$e\vert p^{n}-1$and$N={{p^{n}-1}\over{e}}$. In this paper, we construct a family$S$of$e^{2}N~p$-ary sequences, each member of$S$has period$N$and the magnitudes of correlations of members of$S$are upper bounded by$2\sqrt{p^{n}}=2\sqrt{eN+1}$.

A pair of posets (P,Q ) on [n] is called a weak dual MacWilliams pair (wdMp) if the P-weight enumerator of a linear code uniquely determines the Q-weight enumerator of the dual of that code for every linear code of length n over a finite field. First, we show that (P, P/spl breve/) is a wdMp if and only if the group of all P-weight preserving linear automorphisms of the ambient n-dimensional space over the finite field acts transitively on every P-sphere centered at 0. Here P/spl breve/ is the dual poset of P. Also, we show some equivalent conditions which say that P being weak order poset with Q = P/spl breve/ is essentially the only possible case for (P,Q) to be a wdMp.