Multilevel codes with bounded M-th order running digital sum

Abstract

Summary form only given, as follows. Multilevel sequences with a spectral null of order M (i.e., the power spectral density and its first 2M - 1 derivatives vanish) are characterized by finite-state transition diagrams, whose edge labels satisfy bounds on the variation of the Mth-order running digital sum. Necessary and sufficient conditions for sequences exhibiting a spectral null of order M are given. For this new class of codes a lower bound on the minimum Euclidean distance at the output of partial response channels with spectral null of order P is obtained. It is shown that the distance bound depends on the sum of the orders of code and channel spectral nulls and can be met with equality provided that M + P ≤ 10. The case of M + P > 10 leads to an unsolved problem in number theory. Examples of quaternary codes on dicode channels and their maxentropic power spectral density are given.