Locally Two-weight Property for Linear Codes and Its Application

Abstract

A q-ary linear code is an [n,k,d]_q code, which is a linear code of length n, dimension k and minimum weight d over F_q, the field of order q. A fundamental problem in coding theory is to find n_q(k,d), the minimum length n for which an [n,k,d]_q code exists for given k,d and q. We introduce a new notion "e-locally 2-weight (mod q)" for linear codes over F_q and we give a necessary condition for the property. As an application, we prove the non-existence of some [n,4,d]₉ codes with d ≡ −1 (mod 9), which determines n₉(4,d) for some d.