High-Order Spectral-Null Codes- Constructions and Bounds

Let Y ( n , k ) denote the set of all words of length n over the alphabet { + 1, 11, having a kth order spectral-null at zero frequency. A subset of Y ’ ( n , k ) is a spectral-null code of length n and order k. Upper and lower bounds on the cardinality of 9 ( n , k ) are derived. In particular we prove that ( k 1) log, ( n / k ) I n log, l Y ( n , k)l I log, n ) for infinitely many values of n. On the other hand, we show that Y ( n , k ) is empty unless n is divisible by 2m, where m = [log,k] + 1. Furthermore, bounds on the minimum Hamming distance d of Y ( n , k ) are provided, showing that 2k 5 d I k ( k 1) + 2 for infinitely many n. We also investigate the minimum number of sign changes in a word x EY’(n, k ) and provide an equivalent definition of Y ( 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 log, n + O(log log n ) is presented. Furthermore, we prove that the first nonzero moment of any word in 9 ’ ( n , k ) is divisible by k! and then show how to construct a word with a spectral null of order k whose first nonzero moment is any even multiple of 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 W .