On minimal and quasi-minimal linear codes

Abstract

Minimal linear codes are linear codes such that the support of every codeword does not contain the support of another linearly independent codeword. Such codes have applications in cryptography, e.g. to secret sharing. We here study minimal codes, give new bounds and properties and exhibit families of minimal linear codes. We also introduce and study the notion of quasi-minimal linear codes, which is a relaxation of the notion of minimal linear codes, where two non-zero codewords have the same support if and only if they are linearly dependent. © 2013 Springer-Verlag Berlin Heidelberg.