New Bounds for Generalized Column Distances and Construction of Convolutional Codes

Abstract

Based on known bounds for relative generalized Hamming weights of linear codes, we provide several new bounds for generalized column distances of convolutional codes, including the Griesmer-type bound for generalized column distances. Then we construct several infinite families of convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is small compared with the length of these convolutional codes by using cyclic codes, negacyclic codes and GRS codes. In particular, we obtain some convolutional codes such that the (1, 1)-Griesmer defect of these convolutional codes is zero or one. Next we prove that the 2-generalized column distance sequence $\{d_{2,j}(\mathcal {C})\}_{j=1}^{\infty }$ of any convolutional code $\mathcal {C}$ is increasing and bounded from above, and the limit of the sequence $\{d_{2,j}(\mathcal {C})\}_{j=1}^{\infty }$ is related to the 2-generalized Hamming weight of the convolutional code $\mathcal {C}$ . For $i\ge 3$ , we prove that the i-generalized column distance sequence $\{d_{i,j}(\mathcal {C})\}_{j=\lceil \frac {i}{k}-1\rceil }^{\infty }$ of any convolutional code $\mathcal {C}$ is bounded above and below.