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

The spectral-null code S(n, k) of kth order and length n is the union of n-tuples with ±1 components, having kth-order spectral-null at zero frequency. We determine the exact asymptotic in n behavior of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log₂|S(n, k)|=n-k ²/2log₂n+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 (see ibid., vol40, p.1826-40, 1994)