Asymptotically exact bounds on the size of high-order spectral-null codes

The spectral-null code S(n; k) of k-th order and length n is the union of n-tuples with 1 components, having k-th order spectral null at zero frequency. We determine the exact asymptotic in n behaviour of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log 2 jS(n; k)j = n ? k 2 2 log 2 n + c k + o(1), where c k is a constant depending only on k and o(1) tends to zero when n grows. This is an improvement on the earlier known bounds due to Roth, Siegel and Vardy.