  We develop a spectral method for the spatially homogeneous Boltzmann equation
using Burnett polynomials in the basis functions. Using the sparsity of the
coefficients in the expansion of the collision term, the computational cost is
reduced by one order of magnitude for general collision kernels and by two
orders of magnitude for Maxwell molecules. The proposed method can couple
seamlessly with the BGK-type modelling techniques to make future applications
affordable. The implementation of the algorithm is discussed in detail,
including a numerical scheme to compute all the coefficients accurately, and
the design of the data structure to achieve high cache hit ratio. Numerical
examples are provided to demonstrate the accuracy and efficiency of our method
