High-order spectral-null codes-constructions and bounds

Let 𝒮(n.k) denote the set of all words of length n over the alphabet {+1,-1}, having a k th order spectral-null at zero frequency. A subset of 𝒮(n,k) is a spectral-null code of length n and order k. Upper and lower bounds on the cardinality of 𝒮(n,k) are derived. In particular we prove that (k-1) log₂ (n/k)⩽n-log₂ |𝒮(n,k)|⩽O(2^klog₂n) for infinitely many values of n. On the other hand, we show that 𝒮(n.k) is empty unless n is divisible by 2^m, where m=[log₂k]+1. Furthermore, bounds on the minimum Hamming distance d of 𝒮(n,k) are provided, showing that 2k⩽d⩽k(k-1)+2 for infinitely many n. We also investigate the minimum number of sign changes in a word x∈𝒮(n,k) and provide an equivalent definition of 𝒮(n,k) in terms of the positions of these sign changes. An efficient algorithm for encoding arbitrary information sequences into a second-order spectral-null code of redundancy 3 log₂n+O(log log n) is presented. Furthermore, we prove that the first nonzero moment of any word in 𝒮(n,k) is divisible by k!. This leads to an encoding scheme for spectral-null codes of length n and any fixed order k, with rate approaching unity as n→∞