Satins, lattices, and extended Euclid's algorithm

Motivated by the design of satins with draft of period m and step a, we draw our attention to the lattices $$L(m,a)=\langle (1,a),(0,m)\rangle$$ L ( m , a ) = ⟨ ( 1 , a ) , ( 0 , m ) ⟩ where $$1\le a<m$$ 1 ≤ a < m are integers with $$\gcd (m,a)=1$$ gcd ( m , a ) = 1 . We show that the extended Euclid's algorithm applied to m and a produces a shortest no null vector of L(m, a) and that the algorithm can be used to find an optimal basis of L(m, a). We also analyze square and symmetric satins. For square satins, the extended Euclid's algorithm produces directly the two vectors of an optimal basis. It is known that symmetric satins have either a rectangular or a rombal basis; rectangular basis are optimal, but rombal basis are not always optimal. In both cases, we give the optimal basis directly in terms of m and a.