Systematic study of weakly cyclically permutable codes

Systematically study weakly cyclically permutable codes for multidimensional synchronization and forensic coding, including whether relaxing the requirement of recovering the translation can yield larger codes or simpler constructions.

Background

A cyclically permutable code requires recovery of both the underlying codeword and the translation. The paper defines weakly cyclically permutable codes by relaxing this requirement: only the transmitted codeword must be recovered, so codewords need only belong to distinct translation necklaces.

Because the forensic application itself does not require recovery of the translation, weakly cyclically permutable codes may offer improved code sizes or simpler constructions. The paper explicitly leaves their systematic investigation unresolved.

References

Since all constructions developed in this paper are CPCs, we do not pursue WCPCs further and leave their systematic study for future work.

— Single-Fragment Forensic Coding via Multidimensional Cyclically Permutable Codes  (2609.24575 - Sun et al., 21 Sep 2026) in Remark 2.4, Section 2.2, “Multidimensional CPCs”