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.

Generalized weights of convolutional codes

“…, c k (x) Fq [x] } may not have a minimum with respect to inclusion, as the next example shows. 3 . Then C is a noncatastrophic (3, 2, 2) binary code and 1 + x 2 0 1 1 1 0 and 1 x 2 1 1 1 0 are two row-reduced generator matrices for C whose F q -rowspaces have incomparable supports.…”

“…Let A = (1, 1, 0), (1, 0, 1) F2[x] be the optimal anticode from Example 6.7. It is clear that A is not contained in any subcode of F 2 [x] 3 generated by two vectors of the standard basis of F 3 2 . Moroever, it is easy to show that A ⊥ = (1, 1, 1) F2[x] , in particular A ⊥ is not an elementary optimal anticode.…”

“…Their definition of generalized Hamming weights is the natural extension of the usual definition to infinite dimensional codes. Along the same lines, Cardell, Firer, and Napp in a series of papers [2,4,3] introduce and study a new class of generalized weights for convolutional codes, based on the column distance instead of the Hamming distance.…”