Bent-function sequences

Abstract

In this paper we construct a new family of nonlinear binary signal sets which achieve Welch's lower bound on simultaneous cross correlation and autocorrelation magnitudes. Given a parameter n with n=0 \pmod{4} , the period of the sequences is 2^{n}-1 , the number of sequences in the set is 2^{n/2} , and the cross/auto correlation function has three values with magnitudes \leq 2^{n/2}+1 . The equivalent linear span of the codes is bound above by \sum_{i=1}^{n/4}\left(\stackrel{n}{i} \right) . These new signal sets have the same size and correlation properties as the small set of Kasami codes, but they have important advantages for use in spread spectrum multiple access communications systems. First, the sequences are balances, which represents only a slight advantage. Second, the sequence generators are easy to randomly initialize into any assigned code and hence can be rapidly hopped from sequence to sequence for code division multiple access operation. Most importantly, the codes are nonlinear in that the order of the linear difference equation satisfied by the sequence can be orders of magnitude larger than the number of memory elements in the generator that produced it. This high equivalent linear span assures that the code sequence cannot be readily analyzed by a sophisticated enemy and then used to neutralize the advantages of the spread spectrum processing.