Generalized Column Distances

Abstract

The notion of Generalized Hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography. In this paper we extend this concept to the context of convolutional codes. In particular, we focus on column distances and introduce the novel notion of generalized column distances (GCD). We first show that the hierarchy of GCD is strictly increasing. We then provide characterizations of such distances in terms of the truncated parity-check matrix of the code, that will allow us to determine their values. Finally, the case in which the parity-check matrix is in systematic form is treated.This work was supported in part by the Sao Paulo Research Foundation (FAPESP) under Grant 2013/25977-7. The work of Sara D. Cardell was supported in part by the FAPESP, under Grant 2015/07246-0 and in part by the CAPES. The work of Marcelo Firer was supported in part by the CNPq. The work of Diego Napp was supported in part by the Spanish, Generalitat Valenciana, Univesitat d’Alacant, under Grant AICO/2017/128 and Grant VIGROB-287