‎Let A=[a_₀, a_₁, ..., a_{n-1}] ∈ ℂ^ℓ be a finite complex sequence of length ℓ periodically continued with period ℓ. ‎The periodic autocorrelation function PAF(A,s) of A at lag s is‎ PAF(A, s) =∑ {j=0}^{ℓ-1} a_j\overline{a_{j+s}}, s=0,1, ..., ℓ-1. ‎Two sequences‎ ‎A and B of the same length ℓ. ‎form a Legendre pair (A,B) if‎ ‎PAF(A,s)+PAF(B,s)=-2, s=1,2, …, ⌊n/2⌋. ‎Binary Legendre pairs of length ℓ are pertinent to the construction of Hadamard matrices of order ℓ +2 which can exist only if ℓ is odd‎. ‎It is conjectured that there is a binary Legendre pair of every odd length ℓ which is stronger than the Hadamard-matrix conjecture‎. ‎Several infinite classes of Legendre pairs are known‎. ‎The most prominent constructions are defined via characters of finite fields‎. ‎In particular‎, ‎there are Legendre pairs of length ℓ for every prime ℓ =p and every Mersenne number ℓ =2^s-1‎, ‎respectively‎. ‎The smallest undecided case is ℓ=11. ‎In the first part of the talk we summarize the known results on binary Legendre pairs‎. ‎In the second part we discuss quaternary Legendre pairs of length ℓ‎. ‎In contrast to binary Legendre pairs they can exist for even ℓ as well‎. ‎Very recently‎, ‎the first infinite construction was found by Jedwab and Pender completing our earlier semi-construction‎. ‎We give also constructions of quaternary Legendre pairs of length ℓ for all remaining even ℓ < 42‎. ‎The smallest open case is ℓ =42 and the case ℓ =46 is particularly interesting since the existence of a quaternary Legendre pair of this length would imply the existence of a quaternary Hadamard matrix of order 94. ‎Finally‎, ‎we mention generalizations to k-ary sequences‎.

Zoom room information:
https://us06web.zoom.us/j/84906984159?pwd=BCWaIbXBuku3A5I84zNg9mHFxVZjXD.1
Meeting ID: 849 0698 4159
Passcode: 362880