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We show that the generalized minimum distance function is non-increasing as the degree varies for reduced standard graded algebras over a field. This allows us to define its regularity index and its stabilization value. The stabilization value is computed for every cases. We study how the regularity index varies as the number of polynomial increases, and use this to give bounds for it.

Let $S$ be a polynomial ring over a field $K$, with a monomial order $\prec$, and let $I$ be an unmixed graded ideal of $S$. In this paper we study two functions associated to $I$: the minimum distance function $\delta_I$ and the footprint function ${\rm fp}_I$. It is shown that $\delta_I$ is positive and that ${\rm fp}_I$ is positive if the initial ideal of $I$ is unmixed. Then we show that if $I$ is radical and its associated primes are generated by linear forms, then $\delta_I$ is strictly decreasing until it reaches the asymptotic value $1$. If $I$ is the edge ideal of a Cohen--Macaulay bipartite graph, we show that $\delta_I(d)=1$ for $d$ greater than or equal to the regularity of $S/I$. For a graded ideal of dimension $\geq 1$, whose initial ideal is a complete intersection, we give an exact sharp lower bound for the corresponding minimum distance function., Comment: Commun. Korean Math. Soc., to appear