On de Bruijn Rings and Families of Almost Perfect Maps
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914169492602880 |
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| author | Stelldinger, Peer |
| author_facet | Stelldinger, Peer |
| contents | De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape (m,n) appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd m=n element {3,5,7} and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd m=n element {3,5,7} as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_03309 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On de Bruijn Rings and Families of Almost Perfect Maps Stelldinger, Peer Discrete Mathematics Combinatorics De Bruijn tori, or perfect maps, are two-dimensional periodic arrays of letters from a finite alphabet, where each possible pattern of shape (m,n) appears exactly once in a single period. While the existence of certain de Bruijn tori, such as square tori with odd m=n element {3,5,7} and even alphabet sizes, remains unresolved, sub-perfect maps are often sufficient in applications like positional coding. These maps capture a large number of patterns, with each appearing at most once. While previous methods for generating such sub-perfect maps cover only a fraction of the possible patterns, we present a construction method for generating almost perfect maps for arbitrary pattern shapes and arbitrary non-prime alphabet sizes, including the above mentioned square tori with odd m=n element {3,5,7} as long that the alphabet size is non-prime. This is achieved through the introduction of de Bruijn rings, a minimal-height sub-perfect map and a formalization of the concept of families of almost perfect maps. The generated sub-perfect maps are easily decodable which makes them perfectly suitable for positional coding applications. |
| title | On de Bruijn Rings and Families of Almost Perfect Maps |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2405.03309 |