  Motivated by notions from coding theory, we study the generalized minimum
distance (GMD) function $\delta_I(d,r)$ of a graded ideal $I$ in a polynomial
ring over an arbitrary field using commutative algebraic methods. It is shown
that $\delta_I$ is non-decreasing as a function of $r$ and non-increasing as a
function of $d$. For vanishing ideals over finite fields, we show that
$\delta_I$ is strictly decreasing as a function of $d$ until it stabilizes. We
also study algebraic invariants of Geramita ideals. Those ideals are graded,
unmixed, $1$-dimensional and their associated primes are generated by linear
forms. We also examine GMD functions of complete intersections and show some
special cases of two conjectures of Toh\u{a}neanu--Van Tuyl and
Eisenbud-Green-Harris.
