Punctured Bent Function Sequences for Watermarked DS-CDMA

In this letter, we investigate the effect of inserting some randomly generated watermarking chips into known spreading sequences in terms of periodic correlations; moreover, we give two design criteria for good watermarked sequences in the sense of: 1) reducing the average correlation value and 2) minimizing the variance of correlations. For <inline-formula> <tex-math notation="LaTeX">$n=2m$ </tex-math></inline-formula> with even <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>, we propose a set of <inline-formula> <tex-math notation="LaTeX">$2^{m-1}$ </tex-math></inline-formula> punctured bent function sequences of length <inline-formula> <tex-math notation="LaTeX">$2^{n}-1$ </tex-math></inline-formula> punctured by the Singer difference set. The maximum non-trivial correlation magnitude of the proposed set turns out to be <inline-formula> <tex-math notation="LaTeX">$2^{m}+1$ </tex-math></inline-formula>, which is asymptotically two times the Welch bound.