Further Results on Pure Summing Registers and Complementary Ones

TL;DR

This paper fully characterizes the cycle structures of pure and complementary summing registers, introduces an algorithm to generate de Bruijn cycles from CSR, and discusses limitations in their generalizations.

Contribution

It provides a complete cycle structure analysis of PSR and CSR, and proposes a new algorithm for de Bruijn cycle generation based on CSR.

Findings

Cycle structures of PSR and CSR are fully determined.

An algorithm for de Bruijn cycle generation from CSR is introduced.

Limitations in the generalization of extended representations are identified.

Abstract

We decide completely the cycle structure of pure summing register (PSR) and complementary summing register (CSR). Based on the state diagram of CSR, we derive an algorithm to generate de Bruijn cycles from CSR inspired by Tuvi Etzion's publication in 1984. We then point out the limitation in generalizations of extended representation we use in the algorithm proposed, with a proof of the fact that only PSR and CSR contain pure cycles all dividing n+1.