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Vectorial Boolean functions are used as the nonlinear components in stream ciphers.The security of the cipher system depends on the cryptographic properties of the functions.The widely accepted criteria are balancedness,high nonlinearity,correlation immune and so on.In this paper,we present a new construction method of balanced vectorial Boolean functions with strictly almost optimal nonlinearity.Furthermore.for the first time,the first-order correlation immune functions with currently best known nonlinearity 2n-1-2n/2-1-2[n/4」 are obtained.It is also shown that our functions can have good algebraic degree.