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

The spectral-null code S(n, k) of fcth order and length n is the union of ra-tuples with ±1 components, having fcth-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, Iog2 S(n, k) -n -log2 n + Ck + o(l), where c/, is a constant depending only on k and o(l) tends to zero when n grows. This is an improvement on the earlier known bounds due to Roth, Siegel, and Vardy.