Minimum distance functions of complete intersections

Abstract

Abstract. We study the minimum distance function of a complete intersection graded ideal in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we use the footprint function to give a sharp lower bound for the minimum distance function. Then we show some applications to coding theory.

Steiner systems and configurations of points

“…In particular, we can refer to [48] for some techniques used to study algebraic varieties with special combinatorial features. Recently, connections between commutative algebra and coding theory have gained much attention (see [15,16,19,24,31,34,38,[52][53][54] for more results in this direction). For a background in commutative algebra with a view toward algebraic geometry we suggest [20].…”

Generalized minimum distance functions and algebraic invariants of Geramita ideals

“…Given integers d, r ≥ 1, let M ≺,d,r be the set of all subsets M of ∆ ≺ (I) d := ∆ ≺ (I) ∩ S d with r distinct elements such that (in ≺ (I) : (M )) = in ≺ (I). If r = 1 one obtains the footprint function of I that was studied in [28] from a theoretical point of view (see [24,25] for some applications). In this case we denote fp I (d, 1) simply by fp I (d) and M ≺,d,r by M ≺,d .…”

Steiner systems and configurations of points

“…In particular, we can refer to [52] for some techniques used to study algebraic varieties with special combinatorial features. Recently, connections between commutative algebra and coding theory have gained much attention (see [20,16,17,26,33,36,41,56,57,58] for more results in this direction). For a background in commutative algebra with a view toward algebraic geometry we suggest [22].…”