HIGH-ORDER SPECTRAL-NULL CODES - CONSTRUCTIONS AND BOUNDS

Let T(n,k) denote the set of all words of length n over the alphabet { +1, -1}, having a kth order spectral-null at zero frequency. A subset of T(n,k) is a spectral-null code of length n and order k. Upper and l ower bounds on the cardinality of T(n,k) are derived. In particular we prove that (k - 1) log2 (n/k) less-than-or-equal-to n - log2 \T(n,k)\ less-than-or-equal-to O(2k log2 n) for infinitely many values of n. O n the other hand, we show that T(n,k) is empty unless n is divisible b y 2m, where m = left-perpendicularlog2 kright-perpendicular + 1. Furth ermore, bounds on the minimum Hamming distance d of T(n,k) are provide d, showing that 2k less-than-or-equal-to d less-than-or-equal-to k(k - 1) + 2 for infinitely many n. We also investigate the minimum number of sign changes in a word x is-an-element-of T(n,k) and provide an equ ivalent definition of T(n,k) in terms of the positions of these sign c hanges. An efficient algorithm for encoding arbitrary information sequ ences into a second-order spectral-null code of redundancy 3 log2 n O(log log n) is presented. Furthermore, we prove that the first nonzer o moment of any word in T(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 fo r spectral-null codes of length n and any fixed order k, with rate app roaching unity as n --> infinity.