A note on full weight spectrum codes

TL;DR

This paper characterizes the existence conditions for full weight spectrum linear codes, which contain codewords of all nonzero weights up to their length, using a geometric perspective of generator matrices.

Contribution

It provides necessary and sufficient conditions for the existence of full weight spectrum codes based on geometric properties of generator matrices.

Findings

Derived conditions for FWS code existence

Connected code properties with projective geometry

Enhanced understanding of weight distributions in linear codes

Abstract

A linear $ [n,k]_q $ code $ C $ is said to be a full weight spectrum (FWS) code if there exist codewords of each nonzero weight less than or equal to $ n $. In this brief communication we determine necessary and sufficient conditions for the existence of linear $ [n,k]_q $ full weight spectrum (FWS) codes. Central to our approach is the geometric view of linear codes, whereby columns of a generator matrix correspond to points in $ PG(k-1,q) $.