Families of sequences with optimal Hamming-correlation properties

Algebraic shift register sequences

“…In [121], Lempel and Greenberger showed how to use an m-sequence to construct an optimal family of sequences. Their method has since been applied in a variety of different situations, and it may be described in general terms as follows.…”

“…Provided this happens relatively rarely, an error correcting code can be used to recover from collisions. In their foundational article [121], Lempel and Greenberger defined the Hamming (cross)correlation with shift τ of the sequences a, b to be the number of collisions that occur in a single period, that is,…”

“…In [121] Lempel and Greenberger defined an optimal sequence to be one that meets the bound A Ham (a) ≥ α of Theorem 11.3.7. They defined S to be an optimal family if every sequence a ∈ S is optimal and if every distinct pair of sequences a, b ∈ S meets the bound C Ham (a, b) ≥ α + 1 of Theorem 11.3.7.…”

“…Applying Proposition 14.9.1 gives a family of balanced FH sequences (over the same alphabet F with the same period q m − 1), which contains |V | = q j shift distinct members. This family was originally described in [121].…”

Near Optimal Low-hit-zone Frequency Hopping Sequence Set with Partial Hamming Correlation Property

“…The degree of the mutual interferences closely related to the Hamming crosscorrelation properties of the FHSs ( [1][2][3]). So, it is desirable to employ FHSs with low Hamming correlation to reduce the multiple-access interference of frequencies [4]. Thus, it is an important problem that design FHS set with good Hamming correlation properties.…”