A Weak Structural Form of Commutative Equivalence in Finite Codes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911551138562048 |
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| author | Kraizberg, Dean |
| author_facet | Kraizberg, Dean |
| contents | We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_27656 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Weak Structural Form of Commutative Equivalence in Finite Codes Kraizberg, Dean Information Theory Discrete Mathematics 68R15, 20M05, 94A45 We investigate the structural relationship between prefix-free codes over the binary alphabet and a class of unlabeled rooted trees, which we call \emph{symmetric} trees. We establish a canonical correspondence between prefix-free codes and symmetric trees, preserving not only the lengths of codewords but also some additional commutative structure. Using this correspondence, we provide a result related to the commutative equivalence conjecture. We show that for every code, there exists a prefix-free code such that, for each fixed word length, the sums of powers of two determined by the occurrences of a distinguished symbol are equal. |
| title | A Weak Structural Form of Commutative Equivalence in Finite Codes |
| topic | Information Theory Discrete Mathematics 68R15, 20M05, 94A45 |
| url | https://arxiv.org/abs/2603.27656 |