# On Codes Satisfying Mth-Order Running Digital Sum Constraints

## Abstract

Multilevel sequences with a spectral null of order M at frequency f, meaning that the power spectral density and its first 2M-1 derivatives vanish at f, 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 with a spectral null of order M at dc or an arbitrary rational submultiple of the symbol frequency are given. A multilevel code with bounded Mth-order running digital sum is a subset of the set of all sequences generated by a finite state-transition diagram characterizing sequences with a spectral null of order M. Distance properties of this new class of codes on partial-response channels are examined and a lower bound on the minimum Euclidean distance at the output of partial response channels with a 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. Simple encoders and decoders for selected quaternary codes with a spectral null at dc are given for the dicode channel. The power spectral density of quaternary codes and the maxentropic power spectral density of quaternary sequences with a first-order null are presented. © 1991 IEEE