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 present an analysis on the performance of Plotkin-type UEP codes by using threshold and equivalent channel model under the Gaussian assumption. Specifically, we give a framework on how to assign the rate of each component code of this Plotkin-type UEP code so that the goal of UEP is successfully achieved with far better overall BER performance. The result is surprising in that which component code should be assigned for MSB can be changed according to the assigned rate of each component code and overall rate.

We apply the Hadamard equivalence to all the binary matrices of size m × n and study various properties of this equivalence relation and its classes. We propose to use HR-minimal as a representative of each equivalence class and count the number of HR-minimals of size m × n for m ≤ 3. Some properties and constructions of HR-minimals are investigated. HR-minimals with the largest weight on its second row are defined as Quasi-Hadamard matrices, which are very similar to Hadamard matrices in terms of the absolute correlations of pairs of rows, in the sense that they give a set of row vectors with “best possible orthogonality.” We report lots of exhaustive search results and open problems, one of which is equivalent to the Hadamard conjecture.