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On lattices generated by finite abelian groups. (English) Zbl 1328.11077

Let \(G = \{0,g_1,\dots,g_n\}\) be a finite abelian group. Several authors have previously studied the lattice generated by \(G\), \[ \mathcal{L}(G) = \left\{X = (x_1,\dots,x_n,-x_1-\cdots - x_n) \in \mathbb{Z}^{n+1}: x_1 g_1+\cdots + x_n g_n = 0\right\}. \] In particular, M. Sha [Finite Fields Appl. 31, 84–107 (2015; Zbl 1386.11083)] has shown that when \(G\) is the direct product of two cyclic groups and is not isomorphic to \(\mathbb{Z}/4\mathbb{Z}\), that this lattice has a basis of vectors of minimal length. Sha has also given an upper bound for the covering radius of \(\mathcal{L}(\mathbb{Z}/n\mathbb{Z})\).
The authors of this paper extend this first result on to all finite abelian groups. They find an explicit basis matrix for each cyclic group and then show how these bases can be combined under taking direct products. Several of the proofs rely on computing determinants of particular matrices by applications of the Cauchy-Binet formula. The arguments are very concrete and helpful examples are given.
The authors also improve Sha’s result on the covering radius. The main idea is to first choose a basis matrix and then considering the sublattices spanned by the first \(k\) columns of the matrix. Finally, the authors also express the automorphism group of \(\mathcal{L}(G)\) in terms of the group of automorphisms of \(G\) and give a geometric interpretation of this result.

MSC:

11H31 Lattice packing and covering (number-theoretic aspects)
11G20 Curves over finite and local fields
11H55 Quadratic forms (reduction theory, extreme forms, etc.)
15A15 Determinants, permanents, traces, other special matrix functions
15B05 Toeplitz, Cauchy, and related matrices
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)

Citations:

Zbl 1386.11083
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References:

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