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F-squares over cellular automata
Joanne Hall1, Kerri Morgan1, Stella Stylianou1
1School of Science, RMIT University, Melbourne, Victoria, Australia.
Plos One
|August 12, 2026
Summary
This study explores generating F-squares using cellular automata (CAs). Researchers identified specific local rules that ensure equal row and column sums, leading to F-squares that are either Latin squares or trivial.
Area of Science:
- Discrete Mathematics
- Theoretical Computer Science
- Combinatorics
Background:
- F-squares are arrays crucial for experimental design, characterized by equal element frequency in all rows and columns.
- Cellular automata (CAs) offer a computational framework for generating complex array structures.
Purpose of the Study:
- To investigate the conditions under which cellular automata can generate F-squares.
- To identify specific local rules for CAs that ensure the properties of F-squares.
Main Methods:
- Analysis of row and column sums of arrays generated by CAs over the finite field š¯”½2.
- Focus on binary representations of rows and columns to identify patterns ensuring equal sums.
- Identification and characterization of bipermutive and constant local rules.
Main Results:
- Two families of local rules (bipermutive and constant) were identified that guarantee equal row and column sums.
- The study demonstrates that F-squares generated by CAs over š¯”½2 are exclusively Latin squares or trivial F-squares.
Conclusions:
- Specific local rules in CAs are sufficient for generating F-squares.
- The structure of CA-generated F-squares over š¯”½2 is limited to Latin squares or trivial cases.
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