New parameters of subsets in polynomial schemes

TL;DR

This paper introduces new parameters called zero and dual zero intervals in polynomial schemes, establishing bounds and implications for the structure of subsets and codes within these schemes.

Contribution

It defines zero and dual zero intervals in polynomial schemes and explores their bounds and structural implications for codes and schemes.

Findings

Bounds on lengths of zero and dual zero intervals derived from degree and dual degree.

Large zero or dual zero intervals imply the subset induces a completely regular code or scheme.

Spherical analogue of a dual zero interval is also considered.

Abstract

We define new parameters, a zero interval and a dual zero interval, of subsets in $P$- or $Q$-polynomial schemes. A zero interval of a subset in a $P$-polynomial scheme is a successive interval index for which the inner distribution vanishes, and a dual zero interval of a subset in a $Q$-polynomial scheme is a successive interval index for which the dual inner distribution vanishes. We derive the bounds of the lengths of a zero interval and a dual zero interval using the degree and dual degree respectively, and show that a subset in a $P$-polynomial scheme (resp. a $Q$-polynomial scheme) having a large length of a zero interval (resp. a dual zero interval) induces a completely regular code (resp. a $Q$-polynomial scheme). Moreover, we consider the spherical analogue of a dual zero interval.