New Optimal Binary Sequences with Period 4p4p via Interleaving Ding-Helleseth-Lam Sequences

Binary sequences with optimal autocorrelation play important roles in radar, communication, and cryptography. Finding new binary sequences with optimal autocorrelation has been an interesting research topic in sequence design. Ding-Helleseth-Lam sequences are such a class of binary sequences of period $p$, where $p$ is an odd prime with $p\equiv 1(\bmod~4)$. The objective of this letter is to present a construction of binary sequences of period $4p$ via interleaving four suitable Ding-Helleseth-Lam sequences. This construction generates new binary sequences with optimal autocorrelation which can not be produced by earlier ones

Difference Balanced Functions and Their Generalized Difference Sets

Difference balanced functions from $F_{q^n}^*$ to $F_q$ are closely related to combinatorial designs and naturally define $p$-ary sequences with the ideal two-level autocorrelation. In the literature, all existing such functions are associated with the $d$-homogeneous property, and it was conjectured by Gong and Song that difference balanced functions must be $d$-homogeneous. First we characterize difference balanced functions by generalized difference sets with respect to two exceptional subgroups. We then derive several necessary and sufficient conditions for $d$-homogeneous difference balanced functions. In particular, we reveal an unexpected equivalence between the $d$-homogeneous property and multipliers of generalized difference sets. By determining these multipliers, we prove the Gong-Song conjecture for $q$ prime. Furthermore, we show that every difference balanced function must be balanced or an affine shift of a balanced function.Comment: 17 page