A generalization of Kung's theorem

TL;DR

This paper generalizes Kung's theorem by providing bounds on the minimal number of codewords needed to cover a certain support size in linear codes, using extended weight polynomials.

Contribution

It introduces a new generalization of Kung's theorem relating critical exponents to sums of extended weight polynomials for linear codes.

Findings

Derived upper bounds for the minimal number of codewords covering large supports

Extended the applicability of Kung's theorem to broader code parameters

Connected code support properties with weight polynomial sums

Abstract

We give a generalization of Kung's theorem on critical exponents of linear codes over a finite field, in terms of sums of extended weight polynomials of linear codes. For all i=k+1,...,n, we give an upper bound on the smallest integer m such that there exist m codewords whose union of supports has cardinality at least i.