Let S(N,q) be the set of all words of length N over the bipolar alphabet {-1,+1}, having a q-th order spectral-null at zero frequency. Any subset of S(N,q) is a spectral-null code of length N and order q. We give an equivalent formulation of S(N,q) in terms of codes over the binary alphabet {0,1}. We show that S(N,2) is equivalent to a well known class of single error correcting, all unidirectional error detecting (SEC-AUED) codes. We derive an explicit expression for the redundancy of S(N,2). Further, we give new efficient recursive design methods for second-order spectral-null codes, improving the redundancy of the codes found in the literature.