Given a collection of vectors $V = \{v_1, v_2, ..., v_k \}$ belonging to a lattice $L$ with basis $B$, is there an efficient procedure that can determine whether or not $V$ forms a primitive system for $L$? This means that if $L$ has rank $n \geq k$, you can extend $V$ by adding $n-k$ vectors such that the resulting set is a basis for $L$.
How to check if elements form a primitive system of a lattice
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integer-lattices
1 Answers
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Expand the vectors $v_i$ in terms of the basis vectors, and place their integer coefficients on the columns of a $n \times k$ matrix $\boldsymbol V$. Then the following conditions are equivalent:
$\bullet$ $\boldsymbol V$ is extendable to a basis of $\mathbb{Z}^P$ by the adherence of $P-K$ integer column vectors.
$\bullet$ $\exists$ $\boldsymbol C \in \mathbb{Z}^{K \times P}: \boldsymbol C \boldsymbol V = \mathbb{1}_K$.
$\bullet$ $\gcd_i \{ | \boldsymbol V_i | \} = 1$, where $| \boldsymbol V_i |$ denotes a $K \times K$ minor (determinant of $K \times K$ sub-matrix, formed by picking out $K$ rows of $\boldsymbol V$) and $i$ runs over the set of all $K \times K$ minors, of which there are ${P \choose K}$.
$\bullet$ $\text{HNF}(\boldsymbol V^\top) = \left[ \begin{matrix} \boldsymbol O_{K \times (P-K)} & \mathbb{1}_K \end{matrix}\right]$, where $\text{HNF}(\cdot)$ denotes the Hermite Normal Form. We use the convention that $\text{HNF}(\boldsymbol A)$ is obtained from $\boldsymbol A$ by a sequence of elementary column operations.
This information was gathered from the articles http://ieeexplore.ieee.org/document/1085545/ and https://arxiv.org/pdf/1005.3967.pdf.