Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2 m − 1 for small m (m ≤ 22), is given. As far as we know, for m ∈ {5, 7, 11, 13, 17, 19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m and nonorientable at least for all m ≤ 19. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is also proposed. This invariant applied to APN monomial power permutations gives a classification which coincides with the classification of such permutations via CCZ-equivalence, at least up to m ≤ 17

Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers $\chi_s$ of simplicial complexes naturally generalize the chromatic number $\chi_1$ of a graph. In any fixed dimension $d$, the $s$-chromatic number $\chi_s$ of $d$-complexes can become arbitrarily large for $s\leq\lceil d/2\rceil$ [6,18]. In contrast, $\chi_{d+1}=1$, and only little is known on $\chi_s$ for $\lceil d/2\rceil<s\leq d$. A particular class of $d$-complexes are triangulations of $d$-manifolds. As a consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that $\chi_2$ for surfaces becomes arbitrarily large with growing genus. The proof for this is via Steiner triple systems and is non-constructive. In particular, up to now, no explicit triangulations of surfaces with high $\chi_2$ were known. We show that orientable surfaces of genus at least 20 and non-orientable surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a projective Steiner triple systems, we construct an explicit triangulation of a non-orientable surface of genus 2542 and with face vector $f=(127,8001,5334)$ that has 2-chromatic number 5 or 6. We also give orientable examples with 2-chromatic numbers 5 and 6. For 3-dimensional manifolds, an iterated moment curve construction [18] along with embedding results [6] can be used to produce triangulations with arbitrarily large 2-chromatic number, but of tremendous size. Via a topological version of the geometric construction of [18], we obtain a rather small triangulation of the 3-dimensional sphere $S^3$ with face vector $f=(167,1579,2824,1412)$ and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

Self-embeddings of Hamming Steiner triple systems of small order and APN permutations

The classification, up to isomorphism, of all self-embedding monomial power permutations of Hamming Steiner triple systems of order n = 2 m − 1 for small m (m ≤ 22), is given. As far as we know, for m ∈ {5, 7, 11, 13, 17, 19}, all given self-embeddings in closed surfaces are new. Moreover, they are cyclic for all m and nonorientable at least for all m ≤ 19. For any non prime m, the nonexistence of such self-embeddings in a closed surface is proven. The rotation line spectrum for self-embeddings of Hamming Steiner triple systems in pseudosurfaces with pinch points as an invariant to distinguish APN permutations or, in general, to classify permutations, is also proposed. This invariant applied to APN monomial power permutations gives a classification which coincides with the classification of such permutations via CCZ-equivalence, at least up to m ≤ 17

XXV Congreso Argentino de Ciencias de la Computación - CACIC 2019: libro de actas

Trabajos presentados en el XXV Congreso Argentino de Ciencias de la Computación (CACIC), celebrado en la ciudad de Río Cuarto los días 14 al 18 de octubre de 2019 organizado por la Red de Universidades con Carreras en Informática (RedUNCI) y Facultad de Ciencias Exactas, Físico-Químicas y Naturales - Universidad Nacional de Río CuartoRed de Universidades con Carreras en Informátic