Cumulative distribution function for order 7 de Bruijn weight classes

Abstract

Order n de Bruijn sequences are the period 2n binary sequences from n-stage feedback shift registers. The de Bruijn sequences have good randomness and complexity properties. The quantity of de Bruijn sequences in a weight class of the order n generating functions is an unsolved NP complete problem. Weight class distributions for small n have been obtained by exhaustive searches. This paper uses cumulative distribution function to obtain a high resolution projection of the quantity of de Bruijn sequences in each order 7 weight class. The weight class probability mass function is a shifted Binomial probability mass function which in the limit is accurately represented as a Normal probability density function scaled by a Beta probability density function. The order 7 weight class cumulative distribution function can be modeled as a weighted sum of two Normal cumulative distribution functions.