High-order spectral-null codes-constructions and bounds

Abstract

Let /spl Sscr/(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 /spl Sscr/(n,k) is a spectral-null code of length n and order k. Upper and lower bounds on the cardinality of /spl Sscr/(n,k) are derived. In particular we prove that (k-1) log/sub 2/ (n/k)/spl les/n-log/sub 2/|/spl Sscr/(n,k)|/spl les/O(2/sup k/log/sub 2/n) for infinitely many values of n. On the other hand, we show that /spl Sscr/(n.k) is empty unless n is divisible by 2/sup m/, where m=[log/sub 2/k]+1. Furthermore, bounds on the minimum Hamming distance d of /spl Sscr/(n,k) are provided, showing that 2k/spl les/d/spl les/k(k-1)+2 for infinitely many n. We also investigate the minimum number of sign changes in a word x/spl isin//spl Sscr/(n,k) and provide an equivalent definition of /spl Sscr/(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/sub 2/n+O(log log n) is presented. Furthermore, we prove that the first nonzero moment of any word in /spl Sscr/(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/spl rarr//spl infin/. >