Equitable [[2,10],[6,6]]-partitions of the 12-cube
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866910578866388992 |
|---|---|
| author | Krotov, Denis S. |
| author_facet | Krotov, Denis S. |
| contents | We describe the computer-aided classification of equitable partitions of the $12$-cube with quotient matrix $[[2,10],[6,6]]$, or, equivalently, simple orthogonal arrays OA$(1536,12,2,7)$, or order-$7$ correlation-immune Boolean functions in $12$ variables with $1536$ ones (which completes the classification of unbalanced order-$7$ correlation-immune Boolean functions in $12$ variables). We find that there are $103$ equivalence classes of the considered objects, and there are only two almost-OA$(1536,12,2,8)$ among them. Additionally, we find that there are $40$ equivalence classes of pairs of disjoint simple OA$(1536,12,2,7)$ (equivalently, equitable partitions of the $12$-cube with quotient matrix $[[2,6,4], [6,2,4], [6,6,0]]$) and discuss the existence of a non-simple OA$(1536,12,2,7)$.
Keywords: orthogonal arrays, correlation-immune Boolean functions, equitable partitions, perfect colorings, intriguing sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2012_00038 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Equitable [[2,10],[6,6]]-partitions of the 12-cube Krotov, Denis S. Combinatorics Discrete Mathematics 05B15, 94D10 We describe the computer-aided classification of equitable partitions of the $12$-cube with quotient matrix $[[2,10],[6,6]]$, or, equivalently, simple orthogonal arrays OA$(1536,12,2,7)$, or order-$7$ correlation-immune Boolean functions in $12$ variables with $1536$ ones (which completes the classification of unbalanced order-$7$ correlation-immune Boolean functions in $12$ variables). We find that there are $103$ equivalence classes of the considered objects, and there are only two almost-OA$(1536,12,2,8)$ among them. Additionally, we find that there are $40$ equivalence classes of pairs of disjoint simple OA$(1536,12,2,7)$ (equivalently, equitable partitions of the $12$-cube with quotient matrix $[[2,6,4], [6,2,4], [6,6,0]]$) and discuss the existence of a non-simple OA$(1536,12,2,7)$. Keywords: orthogonal arrays, correlation-immune Boolean functions, equitable partitions, perfect colorings, intriguing sets. |
| title | Equitable [[2,10],[6,6]]-partitions of the 12-cube |
| topic | Combinatorics Discrete Mathematics 05B15, 94D10 |
| url | https://arxiv.org/abs/2012.00038 |