Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$
Mark L. Lewis, Shannon M. Tefft

TL;DR
This paper investigates the possible cycle lengths of sequences generated by a specific Ducci function on finite cyclic groups, identifying conditions that lead to shorter cycles than the maximum possible.
Contribution
It characterizes the range of Ducci cycle lengths on groups and provides conditions for shorter cycles, extending understanding of these sequences' periodic behavior.
Findings
Identifies all possible cycle lengths of Ducci sequences on groups.
Provides criteria for when a sequence's period is less than the maximum.
Analyzes the structure of Ducci cycles in modular settings.
Abstract
Let be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call the Ducci function and the sequence the Ducci sequence of for . Every Ducci sequence enters a cycle, so we can let be the number of tuples in the Ducci cycle of , or the period of . In this paper, we will look at what different possible values of we can have and some conditions that if meets at least one of them, will generate a period smaller than the maximum period.
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Taxonomy
TopicsCoding theory and cryptography · Mathematical Dynamics and Fractals · Analytic Number Theory Research
