On efficient high-order spectral-null codes

Abstract

Let S (N,q) be the set of all words of length N over the bipolar alphabet (-1,+1), having a qth order spectral-null at zero frequency. Any subset of S (N,q) is a spectral-null code of length N and order q. This correspondence gives an equivalent formulation of S(N,q) in terms of codes over the binary alphabet (0,1), shows that S(N,2) is equivalent to a well-known class of single-error correcting and all unidirectional-error detecting (SEC-AUED) codes, derives an explicit expression for the redundancy of S(N,2), and presents new efficient recursive design methods for second-order spectral-null codes which are less redundant than the codes found in the literature.